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This manual documents HydroGeoSphere (HGS) version 2166.

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i

Preface

Effective management of watersheds and ecosystems requires a comprehensive knowledge ofhydrologic processes, and impacts of point-source and non-point source pollution on waterquality. Simulation models are being used increasingly to provide predictive capability insupport of environmental and water resource assessment and restoration projects. However,the models used are often based on simplifications to complex hydrologic and transportprocesses. Such models incorporate restrictive assumptions pertaining to spatial variability,dimensionality and the interactions between various components of the flow and transportprocesses. Realizing the limitations of current models for complex, real-world applications,HydroGeoSphere , a 3-D fully-integrated surface and subsurface flow simulator, wasdeveloped. The HydroGeoSphere model is an outcome of more than two decades ofresearch and development under the leadership of Dr. Edward Sudicky and Dr. PeterForsyth at the University of Waterloo and Dr. Rene Therrien at Laval University. TheHydroGeoSphere code provides a rigorous simulation capability that combines fully-integrated hydrologic/water quality/subsurface flow and transport capabilities with a well-tested set of user-programmable interface tools.

A unique feature in HydroGeoSphere is that when the flow of water is simulated in a fully-integrated mode, water derived from rainfall inputs is allowed to partition into componentssuch as overland and stream flow, evaporation, infiltration, recharge and subsurface dischargeinto surface water features such as lakes, streams and wetlands in a natural, physically-basedfashion. That is, the fully-coupled numerical solution approach allows the simultaneoussolution of both the surface and variably-saturate subsurface flow regimes at each time step.This approach also permits dissolved solutes and thermal energy to be naturally exchangedbetween the surface and subsurface flow domains. The solute and thermal energy transportequations are also solved simultaneously at each timestep in both regimes. This makesHydroGeoSphere a unique and ideal tool to simulate the movement of water, energy andsolutes within watersheds in a realistic, physically-based manner.

HydroGeoSphere uses the control volume finite element approach to simulate coupledsurface and subsurface flow and transport. Fully 3-D simulations of variably-saturatedfractured or granular aquifers may be performed. HydroGeoSphere provides severaldiscretization options ranging from simple rectangular domains to irregular domains withcomplex geometry and layering. Mixed element types provide an efficient mechanismfor simulating flow and transport processes in discrete features such as fractures (2-Drectangular or triangular elements), pumping/injection wells, streams or tile drains (1-Dline elements). External flow stresses can include specified rainfall rates, evapotranspiration,snow melt and aquifer pumping or injection. External transport fluxes can include specifiedconcentrations and mass fluxes, ambient air temperatures and incoming short and long-wavesolar radiation. The HydroGeoSphere model also includes the capability to handle thedissolution of light and dense nonaqueous phase liquids from containminat releases intothe subsurface, as well as the degradation and fate of multiple contaiminat species emittedfrom source zones. HydroGeoSphere includes options for adaptive-time stepping andoutput control procedures and an ILU-preconditioned Krylov subspace sparse iterative

ii

matrix solution package. A Newton-Raphson linearization package provides improvedrobustness. HydroGeoSphere is written in Fortran and is compiled by the Intel VisualFortran® compiler. It will run without modification on any Microsoft Windows® or Linuxbased platforms with sufficient RAM. HydroGeoSphere has also been parallelized usingOpenMP for execution on widely-available shared-memory architectures.

This manual describes the physical and mathematical concepts underlying HydroGeo-Sphere and the implementation of these concepts in the numerical model. It further providesthe user with instructions and guidance on use of the code. Example problems are providedto verify the code and to acquaint the user with its applications.

Contents

1 Introduction 1

1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Integrated Hydrologic Model Conceptualization . . . . . . . . . . . . . . . . 2

1.3 HydroGeoSphere Formulation . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Attributes of HydroGeoSphere . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Operation and Input Options . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Document Organization and Usage Guide . . . . . . . . . . . . . . . . . . . 8

2 Theory 9

2.1 Subsurface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2.1 Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2.2 Discrete Fractures . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2.3 Dual Continuum . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.2.4 1D Hydromechanical coupling . . . . . . . . . . . . . . . . 14

2.2 Surface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16


2.2.2.1 Surface Runoff . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.2.2 Treatment of Rill Storage and Storage Exclusion for Ruraland Urban Environments . . . . . . . . . . . . . . . . . . . 20

2.3 One-Dimensional Hydraulic Features . . . . . . . . . . . . . . . . . . . . . . 22

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CONTENTS iv

2.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22


2.3.2.1 General One-Dimensional Flow . . . . . . . . . . . . . . . . 23

2.3.2.2 Flow Through Subsurface Wells . . . . . . . . . . . . . . . 23

2.3.2.3 Flow Through Subsurface Pipe Systems . . . . . . . . . . . 24

2.3.2.4 Open Channel Flow . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Flow Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.1 Dual Continuum Subsurface Coupling . . . . . . . . . . . . . . . . . 27

2.4.2 Surface - Subsurface Coupling . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.1 Subsurface flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.2 Surface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.3 Interception and Evapotranspiration . . . . . . . . . . . . . . . . . . 28

2.5.3.1 Interception . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.3.2 Evapotranspiration . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Winter Hydrological Processes . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6.1 Surface Water Flow with Snowmelt . . . . . . . . . . . . . . . . . . . 31

2.6.2 Groundwater Flow with Freezing and Thawing of Pore Water . . . . 32

2.6.3 Dual Continuum Freezing and Thawing of Pore Water . . . . . . . . 33

2.7 Solute Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


2.7.1.1 Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . 34

2.7.1.2 Discrete Fractures . . . . . . . . . . . . . . . . . . . . . . . 35

2.7.1.3 Double Porosity . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1.4 Isotopic Fractionation . . . . . . . . . . . . . . . . . . . . . 36

2.7.1.5 Dual Continuum . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1.6 Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.7.1.7 Tile Drains . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7.1.8 Surface runoff . . . . . . . . . . . . . . . . . . . . . . . . . 38

CONTENTS v

2.7.1.9 Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7.1.10 Thermal Energy Transport . . . . . . . . . . . . . . . . . . 39

2.8 Solute Transport Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.8.1 Mobile - Immobile Region Coupling . . . . . . . . . . . . . . . . . . 45

2.8.2 Dual Continuum Subsurface Coupling . . . . . . . . . . . . . . . . . 45

2.8.3 Thermal Energy Coupling . . . . . . . . . . . . . . . . . . . . . . . . 46

2.8.4 Isotopic Fractionation Coupling . . . . . . . . . . . . . . . . . . . . . 47

2.9 Solute Transport Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 47

2.9.1 Subsurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.9.2 Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.10 Travel Time Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.10.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.10.2 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.10.2.1 Forward model . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.10.2.2 Backward model . . . . . . . . . . . . . . . . . . . . . . . . 49

2.10.2.3 Total transit time . . . . . . . . . . . . . . . . . . . . . . . 49

2.10.2.4 Outlet/Inlet Transit Time PDF . . . . . . . . . . . . . . . 50

2.10.2.5 Travel Time Probabilities . . . . . . . . . . . . . . . . . . . 51

2.10.2.6 Age, Life Expectancy and Travel Time Statistics . . . . . . 51

2.10.2.7 Mean Age and Mean Life Expectancy Direct Solutions . . 52

2.10.2.8 Evaluating the travel time PDF from the travel time CDF 53

2.10.2.9 Capture zone probability . . . . . . . . . . . . . . . . . . . 54

3 Numerical Implementation 55

3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2 Control Volume Finite Element Method . . . . . . . . . . . . . . . . . . . . 56

3.3 Discretized Subsurface Flow Equations . . . . . . . . . . . . . . . . . . . . 59

3.3.1 Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.2 Discrete Fractures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.3 Dual Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

CONTENTS vi

3.4 Discretized Surface Flow Equation . . . . . . . . . . . . . . . . . . . . . . . 61


3.6 Flow Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.6.1 Dual-Continuum Subsurface Coupling . . . . . . . . . . . . . . . . . 65

3.6.2 Surface - Subsurface Coupling . . . . . . . . . . . . . . . . . . . . . . 65

3.7 Flow Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.7.1 Subsurface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.7.2 Surface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.7.3 Interception and Evapotranspiration . . . . . . . . . . . . . . . . . . 69

3.7.3.1 Interception . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.7.3.2 Evapotranspiration . . . . . . . . . . . . . . . . . . . . . . 70

3.8 Elemental Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.9 Discretized Solute Transport Equations . . . . . . . . . . . . . . . . . . . . 71

3.9.1 Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.9.2 Discrete Fractures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.9.3 Double Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.9.4 Isotopic Fractionation . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.9.5 Dual Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.9.6 Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.9.7 Tile Drains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.9.8 Surface runoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.9.9 Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.9.10 Thermal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


3.11 Solute Transport Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.12 Solute Transport Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 78

3.12.1 Subsurface Transport . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.13 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.13.1 Matrix Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

CONTENTS vii

3.13.2 Newton-Raphson Method . . . . . . . . . . . . . . . . . . . . . . . . 79

3.13.3 Primary Variable Substitution . . . . . . . . . . . . . . . . . . . . . 81

3.13.4 Time Stepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.13.5 Mass Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.13.6 Solution Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4 Verification Examples 84

4.1 Subsurface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.1.1 Level 1: Drawdown in a Theis Aquifer . . . . . . . . . . . . . . . . . 84

4.1.2 Level 2: Unsaturated Flow Through a Column . . . . . . . . . . . . 85

4.1.3 Level 2: Very Dry Initial Conditions . . . . . . . . . . . . . . . . . . 86

4.1.4 Level 2: Drainage of a Fractured Tuff Column . . . . . . . . . . . . . 88

4.1.5 Level 1: 1-D Hydromechanical Coupling . . . . . . . . . . . . . . . . 94

4.1.6 Level 1: 1-D Hydromechanical Coupling with Externally ComputedStresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.2 Surface Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.1 Level 1: 1-D Surface Flow Study of Govindaraju et al., [1988a and1988b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.2 Level 2: Conjunctive Surface-Subsurface Flow Study of Smith andWoolhiser, [1971] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.3 Level 2: 2-D Surface Flow Study of diGiammarco et al., [1996] . . . 111


4.3.1 Level 1: Groundwater Pumping and Observation Wells . . . . . . . . 111

4.3.2 Level 1: Subsurface Water Supply Systems . . . . . . . . . . . . . . 117

4.3.3 Level 2: Subsurface Drain Systems . . . . . . . . . . . . . . . . . . . 118

4.3.4 Level 1: Subsurface Sewer Systems and Overland Flow . . . . . . . . 119

4.4 Coupled Surface/Subsurface Flow . . . . . . . . . . . . . . . . . . . . . . . . 120

4.4.1 Level 3: 3-D Field Scale Study of Abdul [1985] . . . . . . . . . . . . 120

4.4.2 Level 2: 3-D Surface/Subsurface Flow and Evapotranspiration . . . 125

4.5 Subsurface Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

4.5.1 Level 1: Chain Decay Transport in a Porous Medium . . . . . . . . 125

CONTENTS viii

4.5.2 Level 1: Chain Decay Transport in a Single Fracture . . . . . . . . . 128

4.5.3 Level 1: Time-variable Source Condition . . . . . . . . . . . . . . . . 130

4.5.4 Level 1: Transport in a Dual-Porosity Medium . . . . . . . . . . . . 133

4.5.5 Level 2: Coupled Flow and Transport in a Dual-Permeability Medium 133

4.5.6 Level 2: Transport Due to an Injection/Withdrawal Well . . . . . . 134

4.5.7 Level 1: Two-Dimensional Transport from a Point Source in a SteadyState Uniform Flow Field . . . . . . . . . . . . . . . . . . . . . . . . 137

4.5.8 Level 1: Transport Due to an Injection-Withdrawal Well Pair . . . . 140

4.5.9 Level 2: Two-Dimensional (Areal) Transport of a Contaminant Plumein a Heterogeneous Confined Aquifer with a Pair of Injection AndWithdrawal Wells And Strong Ambient Subsurface Flow . . . . . . . 142

4.5.10 Level 2: Two-Dimensional Transport of a Contaminant Plume in aHeterogeneous Confined Aquifer . . . . . . . . . . . . . . . . . . . . 145

4.5.11 Level 2: Two-Dimensional Transport of Contaminant in the WaterPhase of an Unsaturated Rectangular Soil Slab . . . . . . . . . . . . 146

4.6 Variable-Density Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

4.6.1 Level 2: Variable-Density Flow in Porous Media, Elder’s Problem . . 150

4.6.2 Level 3: Variable-Density Flow in Porous Media, Saltpool Experiment 150

4.6.3 Level 2: Variable-Density Flow in Fractured Porous Media . . . . . 152

4.7 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

4.7.1 Level 1, 2: Heat Transfer in Porous Media . . . . . . . . . . . . . . . 155

4.7.2 Level 1: Heat Transfer in Fractured Media . . . . . . . . . . . . . . . 155

4.7.3 Level 1: Heat Transfer in Fractured Porous Media . . . . . . . . . . 159

4.7.4 Level 2: Heat Transfer in Anisotropic Porous Media . . . . . . . . . 159

4.7.5 Level 2: Borden thermal injection experiment . . . . . . . . . . . . . 163


4.8.1 1D travel time PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5 Illustrative Examples 168


5.1.1 Capture zone probability of a pumping-well . . . . . . . . . . . . . . 168

5.2 Simulating Tidal Fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . 168

CONTENTS ix

References 175

A Mathematical Notation 181

Index 188

List of Figures

1.1 Regional Hydrologic Cycle [Adapted from Viessman and Lewis, 1996]. . . . 3

1.2 Integrated Numerical Simulation of Hydrologic System. . . . . . . . . . . . 4

2.1 Treatment of Storage Terms for Various Settings. . . . . . . . . . . . . . . . 21

2.2 Conceptual Model for Depression Storage and Obstruction Storage Exclusion. 22

2.3 Schematic illustration of a groundwater reservoir Ω, with inlet (Γ−) and outlet(Γ+) boundaries: (a) Age problem with normal flow field; (b) Life expectancyproblem with reversed flow field. The cross stands for a small water sample,to illustrate the random variable total transit time (T ) as the sum of the tworandom variables age (A) and life expectancy (E). . . . . . . . . . . . . . . 50

2.4 Descriptive statistics on the travel time PDF. . . . . . . . . . . . . . . . . . 52

3.1 Spatial Discretization of the Surface Flow System and its Connection to theSubsurface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 Procedure for the evaluation of the travel time PDF at the element centroid,for the case of a brick element. . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3 Schematic illustration of SCPR method. Left: Four quadrangles defining apatch of elements used to evaluate the velocity at a node (red circle), withindicated sample points (+); Right: Example of a patch made of 8 brickelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1 Results for Pumping in a Theis Aquifer. . . . . . . . . . . . . . . . . . . . . 85

4.2 Pressure Head Profiles for the Unsaturated Flow Verification Example. . . . 88

4.3 Schematic for Very Dry Initial Conditions Problem. . . . . . . . . . . . . . 89

4.4 Results For Very Dry Initial Conditions Problem . . . . . . . . . . . . . . . 90

4.5 Verification Example Involving Fractured Porous Tuff, from Wang and Nara-simhan, [1985]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

x

LIST OF FIGURES xi

4.6 Pressure Drop at Selected Points During the Drainage of a Fractured PorousTuff. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.7 Results for 1D Hydromechanical Coupling Example. . . . . . . . . . . . . . 95

4.8 Schematic for 1D Hydromechanical Coupling with External Stresses Example. 97

4.9 Mean Normal Stress Versus Time at Various Elevations. . . . . . . . . . . . 98

4.10 Results for 1D Hydromechanical Coupling with External Stresses Example. 99

4.11 Schematic Description of the Govindaraju et al. [1988a and 1988b] Problem. 100

4.12 Comparison of Normalized Rising Hydrographs for Saint Venant Equations,the Diffusion Wave Approximation [Govindaraju et al., 1988a] and MSVMSfor Fo = 0.5 and K = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.13 Experimental Setup of the Smith and Woolhiser [1971] Study. . . . . . . . . 104

4.14 Moisture Retention Curve for Soil Layer 1. . . . . . . . . . . . . . . . . . . 105

4.15 Hydraulic Conductivity versus Soil Moisture for Soil Layer 1. . . . . . . . . 105

4.16 Soil Saturation Profile at 550 cm from Upstream End for Simulation of theSmith and Woolhiser [1971] Study. . . . . . . . . . . . . . . . . . . . . . . . 107

4.17 Outflow Hydrograph for Simulation of the Smith and Woolhiser [1971] Study. 108

4.18 Surface Water Depth Profiles at Different Times for the Simulation of theSmith and Woolhiser [1971] Study. . . . . . . . . . . . . . . . . . . . . . . . 109

4.19 Fluid Balance Results for the Simulation of the Smith and Woolhiser [1971]Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.20 Schematic Description of 2-D Surface Water Flow Study of diGiammarco etal. [1996]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.21 Outflow Hydrograph for Simulation of 2-D Surface Water Flow Study ofdiGiammarco et al. [1996]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.22 Channel Stage at Outlet for Simulation of 2-D Surface Water Flow Study ofdiGiammarco et al. [1996]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.23 Pumping wells create a cone of depression and aquifer properties can beestimated for confined (a) and unconfined (b) aquifers. A 3-D grid systemshown in (c) was used to simulate the pumping and drawdown relation in acylindrical aquifer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.24 The simulated time-drawdown relations for the common and dual nodeapproaches shown by circles and dashed lines. The results are compared toanalytical solutions represented by the solid line for confined aquifers (a) andunconfined aquifer (b) systems. . . . . . . . . . . . . . . . . . . . . . . . . . 116

LIST OF FIGURES xii

4.25 Steady state results of the numerical and analytical solutions are shownby circles and solid lines. The hydraulic head, axial flux and radial fluxdistributions are plotted along the vertical pumping well axis for confined (a)and unconfined (b) aquifer systems. . . . . . . . . . . . . . . . . . . . . . . 117

4.26 Simulated (symbols) and analytic (lines) solutions for hydraulic head, andaxial flux distributions along a 100-m long pipe of 10-cm diameter, for auniform leakage (a: rectangles and dashed lines) and non-uniform leakage (b:circles and solid lines) cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.27 The vertical profiles at x=15m and x=22m are shown in (a) for the simulatedand the analytical solutions by circles and solid line. The simulated water-table profile, shown in (b), is shown after 15 hours of drainage (triangles) andwere compared to the numerical solutions by Fipps et al. (1986) (solid linewith circles) and MacQuarrie and Sudicky (1996). . . . . . . . . . . . . . . 119

4.28 Simulated water depth and axial flux distributions along a 100-m long 1-mwide rectangular open channel (a). In (b), the simulated depth-flux relationis compared to the simplified analytic relation when ∂h/∂s ≈ ∂z/∂s shownby circles and solid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

4.29 Site Description for Rainfall-Runoff Field Experiment of Abdul [1985] [fromVanderKwaak, 1999]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

4.30 Three-dimensional View of Topography and Finite element Grid, for Simula-tion of the Abdul [1985] Rainfall-Runoff Field Experiment. . . . . . . . . . . 122

4.31 Outflow Hydrograph for Simulation of the Abdul [1985] study. . . . . . . . . 123

4.32 Spatial Distribution of Water Depth after 50 Minutes of Field Experiment for(a) VanderKwaak [1999] and (b) HydroGeoSphere. . . . . . . . . . . . . . 124

4.33 Water Budget Components for the Simulation of the Panday and Huyakorn.[2004]Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

4.34 Results for a 3-member Decay Chain in a Porous Medium at 10000 Years. . 130

4.35 Results for a 3-member Decay Chain in a Fractured Medium at 10000 Years. 131

4.36 Input Function for Time-variable Source Transport. . . . . . . . . . . . . . 131

4.37 Results for a Time-variable Source Function. . . . . . . . . . . . . . . . . . 132

4.38 Results for Transport in a Dual-porosity Medium. . . . . . . . . . . . . . . 134

4.39 Pressure head profiles of the Gerke and Van Genuchten [1993] study. . . . . 136

4.40 Concentration profiles of the Gerke and Van Genuchten [1993] study. . . . . 136

4.41 Injection/withdrawal Well System. . . . . . . . . . . . . . . . . . . . . . . . 137

4.42 Breakthrough Curve from Simulation of Transport Due to an Injection/withdrawalWell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

LIST OF FIGURES xiii

4.43 Two-dimensional Transport from a Point Source. . . . . . . . . . . . . . . . 139

4.44 Concentration Profiles Along the Center Line for t =1400 Days. . . . . . . . 140

4.45 Injection-withdrawal Well Pair. . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.46 Breakthrough Curve of Concentration Solute at the Pumping Well. . . . . . 142

4.47 Problem Description for 2-D Transport in a Heterogeneous Confined Aquifer. 143

4.48 Concentrations Observed at the Pumping Well. . . . . . . . . . . . . . . . . 144

4.49 Mass Budget for the 2-D Transport Simulation. . . . . . . . . . . . . . . . . 144

4.50 Problem Description for 2-D Transport. . . . . . . . . . . . . . . . . . . . . 145

4.51 Hydraulic Head Distribution at the Pumping Well During the Simulation. . 146

4.52 Breakthrough Curve Observed at the Pumping Well for 2-D Transport Simu-lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

4.53 Solute Mass Balance for 2-D Transport Simulation in Transient Flow Field. 147

4.54 Problem Description for 2-D Transport in an Unsaturated Rectangular SoilSlab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.55 Simulated Contaminant Concentrations in an Unsaturated Rectangular SoilSlab at 0.508d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.56 Results of the Elder problem for an extremely fine grid (256 × 128 elementsin the half domain) at 2.5, 5, 10 and 20 years simulation time. Shown are the20%, 40%, 60% and 80% contours. . . . . . . . . . . . . . . . . . . . . . . . 151

4.57 Results of three-dimensional variable-density transport simulations in porousmedia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.58 Variable-density flow in a set vertical fractures embedded in a porous matrix.Shown are the concentration contours 0.1 to 0.9 with a contour interval of 0.4at 2 years simulation time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

4.59 Temperature profiles of 1D heat transfer in an unfractured porous matrix(example 1). Shown are the temperatures in the matrix at 2,148 (left) and4,262 (right) days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

4.60 Temperature profiles of 1D heat transfer in a single fracture within an imper-meable matrix (example 2). Shown are the temperatures in the fracture at2,148 (left) and 4,262 (right) days. . . . . . . . . . . . . . . . . . . . . . . . 158

4.61 Fracture-matrix system used for model verification (Tang et al., 1981). . . . 160

4.62 Temperature profiles of 1D heat transfer in discretely-fractured porous media.Shown are the temperatures in the fracture at 5,000 (left) and 10,000 (right)seconds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

LIST OF FIGURES xiv

4.63 Temperature profiles of 1D heat transfer in discretely-fractured porous media.Shown are the temperatures in the matrix at 10,000 seconds simulation timeat the distances 0.1 (left) and 0.61 (right) m from the fracture. . . . . . . . 162

4.64 The conceptual model for variable-density heat transfer in anisotropic porousmedia (example 4; Yang and Edwards, 2000). The heat source in the vault isdue to the remaining radioactivity of the stored waste. Top and bottom bound-aries are assigned the constant temperatures 6oC and 17.5oC, respectively,with the corresponding geothermal gradient 11.5 K km−1. . . . . . . . . . . 164

4.65 Evolution of temperature in anisotropic porous media with an exponentiallydecreasing heat source. Simulation times are (a) 104 days, (b) 3×105 days,(c) 7×106 days. Shown are isotherms in degrees Celsius. . . . . . . . . . . . 165

4.66 Forward and backward travel time PDF’s versus analytical solution for a 1Dsemi-infinite domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.1 Pumping-well temporal capture zone probability. The aquifer size is 128×128× 32 m. Iso-probability surfaces 0.1-0.5-0.9. . . . . . . . . . . . . . . . 169

5.2 Temporal moment solutions in days: (a) Mean life-expectancy-to-well distri-bution; (b) Mean age; (c) Mean life expectancy; (d) Mean total transit time.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

5.3 Surface domain and mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.4 Tide level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

List of Tables

2.1 Types of Coupling and Dimensionality for Fluid Flow. . . . . . . . . . . . . 27

2.2 Types of Coupling and Dimensionality Solute Transport. . . . . . . . . . . . 44

4.1 Parameter Values for Simulation of Theis Problem. . . . . . . . . . . . . . . 84

4.2 Water Saturation Versus Pressure Head Relationship for the UnsaturatedColumn Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3 Relative Permeability Versus Water Saturation Relationship for the Unsatu-rated Column Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.4 Material Properties for the Simulation of Forsyth et al. [1995], Example 2. . 87

4.5 Parameter Values Used for Wang and Narasimhan [1985] Relationships. . . 91

4.6 Parameter Values for Simulation of 1-D Hydromechanical Coupling Problem. 94

4.7 Parameter Values for Simulation of 1-D Hydromechanical Coupling Problem. 96

4.8 Parameter Values for Simulation of a 1-D Flow Example from Govindaraju etal. [1988a and 1988b]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.9 Parameter Values for Simulation of the Smith and Woolhiser [1971] Experiment.106

4.10 Parameter Values for Simulation of the 3-D Field Scale Study of Abdul [1985]. 121

4.11 Parameter Values for the Simulation of the Panday and Huyakorn[2004] Study.126

4.12 Parameter Values for Chain-decay Transport in a Porous Medium. . . . . . 128

4.13 Parameter Values for Chain-decay Transport in a Fracture. . . . . . . . . . 129

4.14 Parameter Values for Time-varying Source Transport Simulation. . . . . . . 132

4.15 Parameter Values for Dual-porosity Transport Simulation. . . . . . . . . . . 133

4.16 Parameters for the Gerke and Van Genuchten [1993] study. . . . . . . . . . 135

4.17 Model Parameters for the Simulation of Transport From an Injection/extractionWell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

xv

LIST OF TABLES xvi

4.18 Parameters for Simulation of 2-D Transport from a Point Source . . . . . . 139

4.19 Parameters for Simulations of Transport Due to an Injection-withdrawal Pair.141

4.20 Hydraulic Properties of the Rectangular Soil Slab. . . . . . . . . . . . . . . 148

4.21 Physical Parameters Values for Simulation of Transport in an UnsaturatedRectangular Soil Slab. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

4.22 Parameters used for the Saltpool 1 Simulation . . . . . . . . . . . . . . . . . 150

4.23 Parameters used for the Saltpool 1 Simulation . . . . . . . . . . . . . . . . . 152

4.24 Model parameters used in fractured media studies. All parameters are identicalto those used by Shikaze et al. [1998]. . . . . . . . . . . . . . . . . . . . . . 153

4.25 Model parameters used in the verification example for 1D heat transfer inan unfractured porous matrix. All parameters are identical to those used byWard et al. [1984]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

4.26 Model parameters used in the verification example for 2D heat transfer in asingle fracture embedded in a porous matrix. All parameters are identical tothose used by Meyer [2004]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

4.27 Model parameters used in the verification example for 2D variable-densitythermal flow and heat transfer in anisotropic porous media. All parametersare identical to those used by Yang and Edwards [2000]. . . . . . . . . . . . 164

Chapter 1

Introduction

1.1 General

A diverse group of problems exists that requires quantification of the entire hydrologic cycleby integrated simulation of water flow, contaminant and thermal energy migration in thesurface and subsurface regimes. Increased demand on limited resources for potable water andother purposes has driven the development of innovative management practices includingwater recycling, drainage water reuse for salt-tolerant crops, conjunctive use of surface andsubsurface water resources, maintenance of ecosystems and artificial recharge of subsurfaceaquifers during wet periods. A quantification of available water within the hydrologic systemand the impacts of withdrawals is essential for addressing these complex water managementissues. The complex cycle of irrigation, evaporation, infiltration, discharge to nearby lakes,rivers, and streams, and aquifer pumping, needs to be quantified to resolve supply anddemand issues. Concerns over drying and restoration of wetlands or the effects of subsurfacewater withdrawals on surface water features also require an integrated, fully-coupled analysisof the various flow regimes. Ecosystems of lakes, rivers, and wetlands depend on certainminimum flows as do hydropower generation, recreational use, and downstream water users.Regulating water use in hydraulically-connected surface water bodies and surficial aquifersystems necessitates an understanding of surface/subsurface water interactions and overallseasonal hydrologic cycle behavior.

Since the early 1970s, there has been an evolution of hydrologic models for single-eventand continuous simulations of rainfall-runoff processes. Earlier models quantify varioushydrologic components using simplified procedures (including a unit hydrograph method,empirical formulas, system lumping, and analytical equations) that are incapable of describingflow physics and contaminant and thermal energy transport in any detail. In the past,numerical models based on complex multi-dimensional governing equations have not receivedmuch attention because of their computational, distributed input and parameter estimationrequirements. Today, with the availability of powerful personal computers and multi-processorworkstations, efficient computational methods, and sophisticated GIS, remote sensing andadvanced visualization tools, the hydrologic community is realizing the tremendous potential

1

CHAPTER 1. INTRODUCTION 2

and utility of physically-based numerical simulators. As pointed out by Woolhiser [1996,p. 126], “there seems to be little disagreement regarding the usefulness of physically-basedmodels for understanding hydrologic systems.” Models of this type are widely held to offerthe greatest opportunity to examine hydrologic impact of land use change [Refsgaard, 1997;Sharika et al., 2000]. Distributed hydrologic models also have immense potential and utilityfor “forecasting the movement of pollutants and sediments” [Beven, 1985]

The resulting model, which we call HydroGeoSphere, is documented herein. Provided inthe following chapters are detailed descriptions, formulations, verification and applicationexamples, input instructions, output formats and sample data files for the various componentsof the model.

HydroGeoSphere is supported by a flexible, user-friendly programmable interface thatmay be used to seamlessly prepare input data sets, or visualize and interpret simulationresults.

1.2 Integrated Hydrologic Model Conceptualization

HydroGeoSphere is based on a rigorous conceptualization of the hydrologic systemcomprising surface and subsurface flow regimes with interactions. The model is designed totake into account all key components of the hydrologic cycle (Figure 1.1). For each timestep, the model solves surface and subsurface flow, mass and energy transport equationssimultaneously and provides complete water balance and solute budgets. Referring toFigure 1.1, the surface water budget can be written as:

P = (QS2 −QS1)−QGS + ETS +QWS + ∆SS/∆t (1.1)

and the subsurface water budget as:

I = (QG2 −QG1) +QGS + ETG +QWG + ∆SG/∆t (1.2)

giving the total hydrologic budget as the sum of Equations 1.1 and 1.2:

P = (QS2−QS1) + (QG2−QG1) + (ETS +ETG) + (QWS +QWG ) + (∆SS + ∆SG)/∆t (1.3)

where P is the net precipitation (actual precipitation - interception), QS1 and QS2 are thesurface water inflow and outflow, QGS is the surface/subsurface water interactive flow, ETSis the evapotranspiration from the surface flow system, QWS is the overland water withdrawal,∆SS is the surface water storage over time step ∆t, QG1 and QG2 are the subsurface waterinflow and outflow, ETG is the evapotranspiration from the subsurface flow system, QWG isthe subsurface water withdrawal and ∆SG is the subsurface water storage over time step ∆t.

In order to accomplish the integrated analysis, HydroGeoSphere utilizes a rigorous, massconservative modeling approach that fully couples the surface flow and transport equationswith the 3-D, variably-saturated subsurface flow and transport equations. This approachis significantly more robust than previous conjunctive approaches that rely on linkage ofseparate surface and subsurface modeling codes.


Figure 1.1: Regional Hydrologic Cycle [Adapted from Viessman and Lewis, 1996].


Figure 1.2: Integrated Numerical Simulation of Hydrologic System.

1.3 HydroGeoSphere Formulation

HydroGeoSphere uses the two-dimensional, depth-integrated diffusion-wave approxima-tion of the Saint Venant equation for surface water flow [Viessman and Lewis, 1996], therebyneglecting the inertial terms of the momentum equation. An integrated hydrologic analysisis accomplished by the coupled solution of the diffusion-wave equation governing 2-D (areal)surface water flow and the Richards’ equation governing 3-D unsaturated/saturated subsur-face flow. For problems that also involve solute or thermal energy transport, the classicaladvection-dispersion equation is used in all domains.

The surface and subsurface flow and transport domains are discretized simultaneously asshown in Figure 1.2. The gridding options for 2-D surface flow and transport include thedepicted rectangular grid that allows variable grid spacing, as well as a triangular grid optionfor geometric flexibility. The coupled surface and subsurface domains consist of: (1) a singlelayer of surface nodes, shown by triangles in Figure 1.2, situated on the land surface, (2)layers of subsurface soil and aquifer nodes, shown by circles, representing the vadose zone,subsurface aquifers and aquitards, and (3) a set of one-dimensional line elements, shown bysquares, representing surficial channels, wells, tile drains, storm and sanitary drains, watermains or other linear features. The 2-D surface flow grid is draped over the subsurface 3-Dmesh to maintain the areal topography of the land surface and to ensure that the nodes inthe surface grid are coincident with those at the top of the subsurface mesh.


The 3-D saturated-unsaturated flow and transport equations for the vadose and saturatedzones are solved using the control volume finite element method. The top layer of surfacenodes discretizes the 2-D surface flow regime, which is assembled into the matrix equations ina fully-implicit manner using a diffusion-wave approximation to the Saint Venant equations.Two methods are used to couple the two flow and transport domains. The first uses anumerical superposition principle whereby the top layer of nodes represents both surfaceand subsurface domains. The second method uses Darcy flux (for flow) and Fickian (fortransport) relations to transfer water from the surface nodes to the first layer of subsurfacenodes (or vice versa), with the assumption that they are separated by a (possibly) thinboundary layer across which the leakance occurs. A single system of matrix equations arisingfrom both discretized flow and transport regimes is then assembled for the entire hydrologicsetting with appropriate boundary conditions being applied to the combined system. Thefully-integrated set of nonlinear discrete equations is linearized using Newton-Raphsonschemes, and is solved simultaneously in an iterative fashion at every time step.

1.4 Attributes of HydroGeoSphere

HydroGeoSphere is a powerful numerical simulator specifically developed for supportingwater resource and engineering projects pertaining to hydrologic systems with surface andsubsurface flow and transport components.

In terms of simulation capability and computational aspects, the HydroGeoSphere codehas the following attributes:

Fluid flow

• Complete hydrologic cycle modeling using detailed physics of surface and subsurfaceflow in one integrated code. The surface regime can be represented as 2-D areal flow forthe entire surface or as 2-D runoff into 1-D channels. The subsurface regime consistsof 3-D unsaturated/saturated flow. Both regimes naturally interact with each otherthrough considerations of the physics of flow between them.

• Physically-based accounting of all components of the hydrologic cycle water budget.

• Accurate delineation and tracking of the water table position, taking into account flowin the unsaturated zone, delayed yield and vertical flow components.

• Handling of non-ponding or prescribed ponding recharge conditions.

• Automatic handling of seepage face conditions at the land-atmosphere interface.

• Automatic and correct apportioning of the total flow rate of a multi-layer well tothe well nodes, including the simulation of water flow and solute/temperature mixingwithin the water column in the well.

• Accommodation of wellbore storage.


• Arbitrary combinations of porous, discretely-fractured, dual-porosity and dual-permeabilitymedia for the subsurface.

Mass and thermal energy transport

• Capability of modeling non-reactive and reactive chemical species transport in theassociated surface and subsurface flow fields, including solute interactions between thesurface and subsurface flow regimes.

• Calculation of temperatures in the surface and subsurface flow regimes as driven byair temperature and incoming solar radiation, accounting for land surface-atmosphericthermal interactions.

• Accurate handling of fluid and mass/thermal energy exchanges between fractures andmatrix including matrix diffusion effects and solute/thermal energy advection in thematrix.

• Straight or branching decay chains representing degradation reactions.

Numerical methods

• The fully-implicit coupling approach used by the code provides for a robust massconserved solution scheme, which is essential for systems with strong interactionsbetween regimes.

• Parallelization employing OpenMP.

• Advanced computational algorithms and a flexible, user-programmable interface allowthe code to perform unprecedented, fully-integrated, 3-D simulation/animation on apersonal computer.

• Fluid, solute and energy balance tracking.

• Unstructured finite-element grids.

• Axi-symmetric grid option.

• 7-point finite-difference option.

• 8-node block or 6-node prism elements, 4-node tetrahedral elements, 3- and 4-nodeplate elements for fractures and surface water, and 2-node line elements for wells,storm and sanitary sewers, water supply mains and tile drains or other types of linearinfrastructure features.

• Adaptive time-stepping schemes with automatic generation and control of time steps,sub-timing option at select nodes where flow and transport is rapid.

• Straightforward organization and control of simulation output.


• Robust and efficient ILU-preconditioned iterative sparse-matrix solver.

• Robust and efficient Newton-Raphson linearization option.

• Flexible pre- and post-processing capabilities.

For field applications and research investigations, HydroGeoSphere can be used to performevent-based and continuous simulations on widely varying spatial scales ranging from singlesoil column profiles to large-scale basins, which may include several catchments. Examplesof field applications of HydroGeoSphere include:

• Integrated water resource assessment.

• Watershed hydrologic analysis, including impacts of land-use or climate-change impactson both surface and subsurface water.

• Floodplain hydrologic analysis.

• Fluvial hydraulic analysis.

• Contaminant migration and fate in both surface and subsurface water.

1.5 Operation and Input Options

The HydroGeoSphere conjunctive surface-subsurface flow simulator enjoys the benefit ofhaving already available and affordable GUI tools for grid generation and subsurface flowmodel input as well as Tecplot for 3-D visualization and animation. In order to handlespatial data analysis and visualization of surface water domain, GIS tools such as ArcViewand ArcInfo may be used.

The modular code features and input/output structures of HydroGeoSphere result infour steps to solve a given problem.

1. Build the necessary data files for the pre-processor grok (as discussed in the referencemanual).

2. Run grok to generate the input data files for HydroGeoSphere.

3. Run HydroGeoSphere to solve the problem and generate output data files.

4. Postprocess the output files to visualize and analyze the results and produce reports.

At a minimum, Step 1 involves creating data files that contain information for discretizing theproblem domain, defining material properties for each element and specifying flow boundaryconditions.

Simulation output pertaining to surface flow regime calculations is reported to the mainHydroGeoSphere output file in a similar manner to that for the subsurface flow calculations.Binary files are created individually for further investigation or post processing.


1.6 Document Organization and Usage Guide

This document is organized into 5 chapters. These chapters and their purposes are outlinedbelow.

Chapter 2 presents the mathematical theory that describes the various physical processesthat are presented in the model.

Chapter 3 shows how the mathematical theory is implemented in the HydroGeoSpherecode, with an emphasis on the numerical techniques used to address non-linearities thatarise when dealing with, for example, variably-saturated flow.

Chapter 4 contains descriptions of several verification problems that were designed to testand demonstrate the capabilities of HydroGeoSphere in solving a variety of flow andtransport phenomena.

The reference manual introduces the user to the basic operations of the pre-processor anddescribes the input instructions used to determine:

• Problem description.

• Simulation control.

• Grid generation.

• Selection of grid components (nodes, elements, etc).

as well as for the saturated-unsaturated subsurface flow, surface flow, and solute and energytransport problems.

A complete list of references cited in each chapter can be found in Chapter 5.2.

Mathematical notation is summarized in Appendix A.

A comprehensive index can be found at the end of this document.

Chapter 2

Theory

2.1 Subsurface Flow

2.1.1 General

The current implementation of HydroGeoSphere assumes that the subsurface flow equationin a porous medium is always solved during a simulation, either for fully-saturated or variably-saturated flow conditions. We therefore first present the basic subsurface flow equationsolved by HydroGeoSphere. This basic equation can be expanded to incorporate, amongother features, discrete fractures, a second interacting porous continuum (e.g. fractures ormacropores), wells, tile drains and surface flow. The following assumptions are made forsubsurface flow:

• The fluid is essentially incompressible.

• The porous medium and fractures (or macropores), if present, are non-deformable.

• The system is under isothermal conditions.

• The air phase is infinitely mobile.

2.1.2 Governing Equations

2.1.2.1 Porous Medium

The following modified form of Richards’ equation is used to describe three-dimensionaltransient subsurface flow in a variably-saturated porous medium:

−∇ · (wmq) +∑

Γex ±Q = wm∂

∂t(θsSw) (2.1)

where wm [dimensionless] is the volumetric fraction of the total porosity occupied by theporous medium (or primary continuum). This volumetric fraction is always equal to 1.0

9

CHAPTER 2. THEORY 10

except when a second porous continuum is considered for a simulation, which is the casewhen the dual continuum option is used to represent existing fractures or macropores. Thefluid flux q [L T−1] is given by:

q = −K · kr∇(ψ + z) (2.2)

where kr = kr(Sw) represents the relative permeability of the medium [dimensionless] withrespect to the degree of water saturation Sw [dimensionless], ψ is the pressure head [L],z is the elevation head [L] and θs is the saturated water content [dimensionless], which isassumed equal to the porosity. Fluid exchange with the outside of the simulation domain, asspecified from boundary conditions, is represented by Q [L3 L−3 T−1], which is a volumetricfluid flux per unit volume representing a source (positive) or a sink (negative) to the porousmedium system.

The hydraulic conductivity tensor, K [L T−1], is given by:

K = ρg

µk (2.3)

where g is the gravitational acceleration [L T−2], µ is the viscosity of water [M L−1 T−1],k is the permeability tensor of the porous medium [L2] and ρ is the density of water [ML−3], which can be a function of the concentration C [M L−3] of any given solute such thatρ = ρ(C).

Water saturation is related to the water content θ [dimensionless] according to:

Sw = θ

θs(2.4)

In Equation 2.1, Γex represents the volumetric fluid exchange rate [L3 L−3 T−1] between thesubsurface domain and all other types of domains supported by the model and it is expressedper unit volume of the other domain types. Currently, these additional domains are surface,wells, tile drains, discrete fractures and dual continuum. The definition of Γex (positive forflow into the porous medium) depends on the conceptualization of fluid exchange betweenthe domains and will be defined in later sections that discuss these respective flow domains.In the equations shown for the other domains, we will use the notation ex=f, ex=d, ex=w,ex=t, ex=o, ex=c, for the fracture, dual continuum, well, tile drain, surface and channeldomains, respectively.

The primary variable of solution for the nonlinear flow Equation 2.1 is the pressure head,and constitutive relations must be established that relate the primary unknown ψ to thesecondary variables Sw and kr. The relative permeability may be expressed in terms ofeither the pressure head or the water saturation. Commonly used functions incorporated inthe model are those presented by Van Genuchten [1980] and Brooks and Corey [1964].

Using the Brooks and Corey functions, the saturation is given by:

Sw = Swr + (1− Swr) | αψ |−β for ψ < −1/αSw = 1 for ψ ≥ −1/α (2.5)


and the relative permeability is obtained from:

kr = Se(2/β+lp+2) (2.6)

where α [L−1] is the inverse of the air-entry pressure head, β [-] is the pore-size distributionindex, lp is the pore-connectivity parameter assumed equal to 2.0 in Brooks and Corey [1964],and Se is often called an effective saturation, given by Se = (Sw − Swr)/(1− Swr), with Swrbeing the residual water saturation [-].

Based on earlier work by Mualem [1976], Van Genuchten [1980] later proposed the followingsaturation-pressure relation:

Sw = Swr + (1− Swr)[1+ | αψ |β

]−νfor ψ < 0

Sw = 1 for ψ ≥ 0(2.7)

with the relative permeability given by:

kr = S(lp)e

[1−

(1− S1/ν

e

)ν]2(2.8)

where (ν = 1− 1

β

), β > 1 (2.9)

and where α and β are usually obtained from a fit of Equations 2.7 and 2.8 to experimentalresults. For his relative permeability model, Mualem [1976] has estimated that the pore-connectivity parameter lp is equal to 0.5 for most soils.

Although other fundamental relations exist, any arbitrary, but physically realistic, functionfor Sw(ψ) and kr(Se) can also be handled through the use of tabular data input for theseparameters, which is also made available in the model. Furthermore, hysteresis is notconsidered here, but may be included in a future version of HydroGeoSphere.

To describe subsurface flow in the saturated zone, the storage term forming the right-handside of Equation 2.1 is expanded in a way similar to that presented by Cooley [1971] andNeuman [1973]. They relate a change in storage in the saturated zone to a change influid pressure through compressibility terms as is conventionally done in hydrogeologicalapplications. They also assume that the bulk compressibility of the medium is constant forsaturated and nearly-saturated conditions. For unsaturated conditions, it is assumed thatthe compressibility effects on storage of water are negligible compared to the effect of changesin saturation. The following expression is obtained for the storage term in Equation 2.1[Cooley, 1971; Neuman, 1973]:

∂

∂t(θsSw) ≈ SwSs

∂ψ

∂t+ θs

∂Sw∂t

(2.10)

where Ss is the specific storage coefficient of the porous medium [L−1].


2.1.2.2 Discrete Fractures

A fracture is idealized here as the space between two-dimensional parallel surfaces, with thetacit assumptions that the total head is uniform across the fracture width. The equationfor variably-saturated flow in a fracture of width wf [L] can be written by extending thesaturated fracture flow equations [Berkowitz et al., 1988; Sudicky and McLaren, 1992] andusing the analogy of Richards Equation (Equation 2.1) for the porous matrix. With thisextension, the governing two-dimensional flow equation in a fracture has the form:

−∇ · (wfqf )− wfΓf = wf∂Swf∂t

(2.11)

where the fluid flux qf [L T−1] is given by:

qf = −Kf · krf∇(ψf + zf ) (2.12)

where ∇ is the two-dimensional gradient operator defined in the fracture plane, krf is therelative permeability of the fracture [-], ψf and zf are the pressure and the elevation headswithin the fracture [L], and Swf is the water saturation for the fracture [-]. The saturatedhydraulic conductivity of a fracture Kf [L T−1], having a uniform aperture wf is given by[Bear, 1972]:

Kf =ρg w2

f

12µ (2.13)

where the fluid density can be a function of the concentration Cf of any given solute in thefracture [M L−3], such that ρ = ρ(Cf ).

Constitutive relations are also required to describe variably-saturated flow in the fractures.There is a very limited number of studies where these relationships have been derivedexperimentally [e.g. Reitsma and Kueper, 1994]. Several theoretical studies have, nevertheless,been performed in order to characterize the nature of the relationships. Wang and Narasimhan[1985] and Rasmussen and Evans [1988] generated synthetic relations between pressure,saturation and relative permeability for a single fracture surface containing a distributionof apertures. Their results were based on capillary theory and they used the well-knowncubic law to represent flow in the fracture. Pruess and Tsang [1990] considered the problemof two-phase flow in a rough-walled fracture surface. They subdivided the fracture surfaceinto sub-elements and assigned a spatially-correlated aperture to each. The occupancy ofeach element by either the wetting or the non-wetting phase fluid was based on a prescribedentry pressure relationship. Once the fluid occupancy was assigned to each element, therelative permeabilities were obtained by performing two single-phase flow simulations: onefor the region occupied by the wetting phase and the other for the non-wetting phase.

In HydroGeoSphere, relative permeability and saturation-pressure head relationships forthe fractures are given by either the Brooks-Corey model (Equations 2.5 and 2.6), the VanGenuchten model (Equations 2.7 and 2.8) or they can be given in tabular forms, which givesflexibility to the user and does not restrict data entry to fixed functions.

Another process that could be considered when describing flow in variably-saturated fracturedporous media is the reduction of the area available for flow across a fracture-matrix interface.


Some portions of a fracture, as it desaturates, cannot transmit any water and thus reducethe area available for matrix flow across the fracture. We use the approach of Wang andNarasimhan [1985] who represented this phenomenon with a function describing the changein effective fracture-matrix area as a function of pressure. This effective area is only appliedto those matrix nodes that are also fracture nodes.

Using arguments similar to those invoked for the porous medium equation, the storage termin Equation 2.11 describing variably-saturated flow in the fractures becomes:

∂

∂tSwf ≈ SwfSsf

∂ψf∂t

+ ∂Swf∂t

(2.14)

where Ssf is the specific storage coefficient for the fractures [L−1]. Because it is assumed herethat the fractures are non-deformable and fluid-filled, there is no contribution to the storageterm from fracture compressibility. Thus, the specific storage coefficient for a fracture undersaturated conditions is related to the water compressibility, αw [L T2 M−1], according to:

Ssf = ρgαw (2.15)

The validity of assuming non-deformable fractures is likely to be reasonable if the fractureshave a high normal stiffness or if changes in the effective stress field within the system dueto pumping, for example, are small.

2.1.2.3 Dual Continuum

The HydroGeoSphere model can simulate variably-saturated fluid flow in a dual continuumbased on the formulation presented by Gerke and Van Genuchten [1993]. The dual continuumformulation in HydroGeoSphere involves 2 separate continua, with the first continuumrepresented by the porous medium. We present here the formulation for the second continuum,which is linked to the porous medium continuum by a fluid exchange term. This secondcontinuum could represent, for example, fractures or macropores that are present in a porousmatrix.

Similarly to flow in the porous medium, three-dimensional variably-saturated flow in thesecond continuum is described by a modified form of Richards’ equation:

−∇ · (wdqd)− Γd ±Qd = wd∂

∂t(θsdSwd) (2.16)

where the fluid flux qd [L T−1] is given by:

qd = −Kd · krd∇(ψd + zd) (2.17)

where krd is the relative permeability of the medium [-] with respect to the degree of watersaturation Swd [-], ψd is the pressure head [L], zd is the elevation head [L] and θsd is thesaturated water content [-], which is equal to the porosity of the dual continuum. Fluidexchange with the outside of the simulation domain is represented by a volumetric fluid fluxper unit volume Qd [L3 L−3 T−1]. The volumetric fraction of the total porosity occupied by


the dual continuum is given by wd [-]. We assume here that the sum of volumetric fractionof the dual continuum wd and that of the porous medium wm is equal to 1.0.

The hydraulic conductivity tensor of the dual continuum, Kd [L T−1], is given by:

Kd = ρg

µkd (2.18)

where kd is the permeability tensor of the dual continuum [L2] and where the density ofwater can be a function of the concentration Cd [M L−3] of any given solute in the dualcontinuum such that ρ = ρ(Cd).

Similarly to the porous medium, the functional relationships relating pressure head tosaturation and relative permeability to saturation are described by either the Van Genuchtenor the Brooks-Corey functions, or are given in tabular form.

For cases where the dual continuum represents fractures, some expressions can be derived torelate the permeability of the fractures to a representative fracture aperture and spacing(for example, Bear 1972). For example, for a set of parallel fractures of uniform aperture wfwith a uniform spacing equal to fs [L], and assuming one-dimensional flow in a directionparallel to the fracture, the equivalent permeability of the set of fractures is given by:

kd =w2f

12 with wd = wffs

(2.19)

For the second continuum the storage term in Equation 2.16 is expanded in a manner similarto Equation 2.10 for the porous medium, using the specific storage Ssd [L−1] for the secondcontinuum.

2.1.2.4 1D Hydromechanical coupling

The HydroGeoSphere code can be used to simulate transient flow affected by surfaceloading conditions. Examples of surface loading conditions include: advent or retreat ofglaciation, erosion, and deposition.

Under hydromechanical equilibrium conditions, the governing equations are [Neuzil, 2003]:

∂σij∂xj

= 0 (2.20a)

σij = 2Gεij + 2G ν

1− 2ν εkkδij + αpδij (2.20b)

∂

∂xi

(Kij

∂h

∂xj

)= Ss3

∂h

∂t− βSs3

1ρg

∂σt∂t±Q (2.21)

where σij is the total stress [M T−2 L−1] defined by Equation 2.20b, εij is the mechanicalstrain [-], G is the shear modulus [M T−2 L−1], ν is Poisson’s ratio [-], α is the effective-stress


hydroelastic constant [-], p is fluid pressure [M T−2 L−1], Ss3 is the three-dimensional specificstorage coefficient [L−1] of the porous medium originally defined in Equation 2.10, β is thethree-dimensional loading efficiency [-], and σt is the mean normal total stress [M T−2 L−1].The loading efficiency is the fraction of stress that is transferred to fluid pressure. The ratioσt/ρg may be interpreted as equivalent fresh water height of the mean normal total stressat a given point. The three-dimensional loading efficiency, α, shear modulus, mean normaltotal stress, and specific storage are defined below:

β =

(1K −

1Ks

)(

1K −

1Ks

)+(θsKf− θs

Ks

) (2.22)

α =(

1− K

Ks

)(2.23)

G = E

2 (1 + ν) (2.24)

σt = σkk3 (2.25)

Ss3 =( 1K− 1Ks

)+ ρg

(θsKf− θsKs

)(2.26)

where K is the bulk modulus of porous media [M T−2 L−1], Ks is the solid bulk modulus [MT−2 L−1], Kf is the fluid bulk modulus [M T−2 L−1], and E is Young’s elastic modulus [MT−2 L−1]. The three-dimensional loading efficiency is the ratio of change in fluid pressure tochange in mean total stress under undrained conditions. For highly compressible media, βapproaches unity while in stiff media it can be zero [Neuzil, 2003]. As porosity approacheszero, K approaches Ks and the effects of water pressure on total stress (Equation 2.20b) andthe effects of change of total stress on water storage (Equations 2.22 and 2.26) also vanish.

However, the current version of HydroGeoSphere does not have a mechanical or anequilibrium module that is coupled with the flow module. The mean normal total stress,which may be spatially distributed and vary with time, must be estimated externallyto HydroGeoSphere. However, under the condition of purely vertical strain, the flowcomponent can be decoupled or partially decoupled from the mechanical component.

Mechanical and hydraulic behaviour can also be partially coupled under the conditionsof relatively homogeneous and areally extensive vertical loading. Under these conditions,Equation 2.21 is written as

∂

∂xi

(Kij

∂h

∂xj

)= Ss1

∂h

∂t− ζSs1

1ρg

∂σzz∂t±Q (2.27)

where σzz is the vertical total stress [M T−2 L−1], Ss1 is the modified one-dimensionalspecific storage [L−1] [Neuzil, 2003] and ζ is the one-dimensional loading efficiency [-]. Ss1and ζ are defined below:

Ss1 =( 1K− 1Ks

)(1− 2α 1− 2ν

3(1− ν)

)+ ρg

(θsKf− θsKs

)(2.28a)


ζ = β(1 + ν)3(1− ν)− 2αβ(1− 2ν) (2.28b)

It can be seen that Equation 2.27 is a subset of Equation 2.21 which is more general.

Equations 2.20 to 2.26 are developed for isotropic materials, for transversely anisotropicmaterials (isotropic along horizontal planes and anisotropic in the vertical direction), β , σt,and Ss3 in Equation 2.21 must be replaced by:

β′ =1KH− 1

Ks(2

3KH− 1

3KV− 1

Ks

)+(θsKf− θs

Ks

) (2.29)

σ′t = σxx3 + σyy

3 + σzz3

( 1KV− 1

Ks

1KH− 1

Ks

)(2.30)

S′s3 =( 2

3KH+ 1

3KV− 1Ks

)+(θsKf− θsKs

)(2.31)

where σxx, and σyy are total stresses [M T−2 L−1] in the horizontal x and y directions,respectively, σzz is the total stress [M T−2 L−1] in the vertical direction, KH and KV arebulk moduli in the horizontal and vertical directions, respectively. It is implicitly assumedthat solid grain compressibility is isotropic. A general formulation for anisotropic materialsis given by Guvanasen and Chan [2000].

2.2 Surface Flow

2.2.1 General

This section describes the mathematical theory of the surface water flow package of theHydroGeoSphere simulator. Surface flow on catchment basins is an important componentof the hydrologic cycle, governing flow to and from the subsurface, channel networks, rivers,lakes and reservoirs. Lake and reservoir flow dynamics and hydrologic conditions of wetlandsare also areal surface flow processes.

Areal surface flow is represented in HydroGeoSphere by a two-dimensional depth-averagedflow equation, which is the diffusion-wave approximation of the Saint Venant equation for sur-face water flow. Before presenting the diffusion-wave equation solved by HydroGeoSpherewe present the simplifications needed to obtain this equation from the full two-dimensionalSaint Venant equations.



2.2.2.1 Surface Runoff

The two-dimensional Saint Venant equations for unsteady shallow water flow consist of 3equations, which are given by the following mass balance equation:

∂φoho∂t

+ ∂(vxodo)∂x

+ ∂(vyodo)∂y

+ doΓo ±Qo = 0 (2.32)

coupled with the momentum equation for the x-direction:

∂

∂t(vxodo) + ∂

∂x(v2xodo) + ∂

∂y(vxovyodo) + gdo

∂do∂x

= gdo(Sox − Sfx) (2.33)

and the momentum equation for the y-direction:

∂

∂t(vyodo) + ∂

∂y(v2yodo) + ∂

∂x(vxovyodo) + gdo

∂do∂y

= gdo(Soy − Sfy) (2.34)

where do is the depth of flow [L], zo is the bed (land surface) elevation [L], ho is the watersurface elevation [L] (ho = zo + do), vxo and vyo are the vertically averaged flow velocities inthe x− and y−directions [L T−1], Qo is a volumetric flow rate per unit area representingexternal source and sinks [L T−1], and φo is a surface flow domain porosity which is unity forflow over a flat plane, and varies between zero at the land surface and unity at the top of allrills and obstructions, for flow over an uneven surface. This conceptualization is discussedfurther in Section 2.2.2.2.

The variables Sox, Soy, Sfx, and Sfy are dimensionless bed and friction slopes in the x− andy−directions, respectively. These slopes can be approximated with either the Manning, theChezy or the Darcy-Weisbach equations. Using the Manning equation, the friction slopesare approximated by:

Sfx = vxovson2x

d4/3o

(2.35)

and:Sfy =

vyovson2y

d4/3o

(2.36)

where vso is the vertically averaged velocity [L T−1] along the direction of maximum slopes (vso =

√v2xo + v2

yo), nx and ny are the Manning roughness coefficients [L−1/3 T] in the xand y directions.

Using the Chezy equation, the friction slopes are approximated by:

Sfx = 1C2x

vxovsodo

(2.35)

and:Sfy = 1

C2y

vyovsodo

(2.36)


where Cx and Cy are the Chezy coefficients [L1/2 T−1] in the x and y directions.

Using the Darcy-Weisbach equation, the friction slopes are approximated by:

Sfx = fx8gvxovsodo

(2.35)

and:Sfy = fy

8gvyovsodo

(2.36)

where fx and fy are dimensionless Darcy-Weisbach friction factors in the x and y directions.The friction factors fx and fy may be obtained from a Moody diagram which can beapproximated for laminar flow as [Akan and Yen, 1981]:

fi = CLRei

(2.37)

where the index i represents the x or y direction, CL is a constant which depends on rainfallintensity as:

CL = 24 + 27.162r0.407 (2.38)

where r is the rainfall intensity in inches hour−1, and Rei is the Reynolds number in directioni given as:

Rei = viodoγ

(2.39)

where γ is the kinematic viscosity [L2 T−1].

Momentum Equations 2.33 and 2.34 can be simplified by neglecting the first three termson the left hand side, representing inertia [Gottardi and Venutelli, 1993] and using eitherapproximation of the friction slopes (Equations 2.35 and 2.36) to give:

vox = −Kox∂ho∂x

(2.40)

and:voy = −Koy

∂ho∂y

(2.41)

where Kox and Koy are surface conductances [L T−1] that depend on the equation used toapproximate the friction slopes. Conductances for the Manning equation are given by:

Kox = d2/3o

nx

1[∂ho/∂s]1/2

(2.42)

and:

Koy = d2/3o

ny

1[∂ho/∂s]1/2

(2.43)

where s is taken in the direction of maximum slope.


For the Chezy equation, Kox and Koy are given by:

Kox = Cxd1/2o

1[∂ho/∂s]1/2

(2.42)

and:Koy = Cyd

1/2o

1[∂ho/∂s]1/2

(2.43)

Similarly, for the Darcy-Weisbach relation, Kox and Koy are given by:

Kox =√

8gfx

1[∂ho/∂s]1/2

(2.42)

and:

Koy =√

8gfx

1[∂ho/∂s]1/2

(2.43)

Comparison of the various expressions for the conductance indicates the following relationshipbetween the coefficients of the Manning, Chezy and Darcy-Weisbach equations:

Cx = d1/6o

nx=√

8gfx

and Cy = d1/6o

ny=√

8gfy

(2.44)

The surface flow equation solved by HydroGeoSphere is finally obtained by substitutingEquations 2.40 and 2.41 into continuity Equation 2.32, which gives the following diffusionwave approximation for surface flow:

∂φoho∂t

− ∂

∂x

(doKox

∂ho∂x

)− ∂

∂y

(doKoy

∂ho∂y

)+ doΓo ±Qo = 0 (2.45)

with Equations 2.42 and 2.43 providing expressions for conductances Kox and Koy.

In addition to neglecting the inertial terms, the assumptions associated with the diffusion-wave equation are those of the Saint Venant equations, which are depth-averaged flowvelocities, hydrostatic pressure distribution vertically, mild slope, and dominant bottomshear stresses. Furthermore, it is assumed that Manning’s, Chezy’s, or Darcy-Weisbach’sformula are valid to calculate frictional resistance forces for unsteady flow.

To simplify the presentation of the discretized surface flow equation in the next chapter,Equation 2.45 is rewritten in vectorial notation:

−∇ · (doqo)− doΓo ±Qo = ∂φoho∂t

(2.46)

where the fluid flux qo [L T−1] is given by:

qo = −Ko · kro∇(do + zo) (2.47)

where kro is a dimensionless factor that accounts for the reduction in horizontal conductancefrom obstruction storage exclusion, which is discussed in Section 2.2.2.2.


2.2.2.2 Treatment of Rill Storage and Storage Exclusion for Rural and UrbanEnvironments

For investigations of urban runoff or agricultural settings, the storage and flow terms ofEquation 2.45 have been further enhanced to include rill storage, also known as depressionstorage, and exclusion effects from obstruction storage. Figure 2.1 shows various physicalsettings that can be simulated by the model, given an appropriate formulation of the storageand flow terms.

It can be assumed that flow occurs over a flat plane, as shown in Figure 2.1a, without anyrill storage or obstruction storage effects. In that case, the surface flow domain porosityφo is always unity. For flood calculations in urban environments, the setting may be muchdifferent as shown in Figure 2.1b, with flow occurring between the grid of buildings in anaveraged sense over the grid area. If the flood is high enough to cover the buildings, onlythen is the full area available for flow and storage of water, otherwise, if not accounted for,lower flood-depths and incorrect discharge would be predicted for an urban flood event. Thestorage capacity that is reduced by the presence of the urban features is called obstructionstorage exclusion. Obstruction storage exclusion may also result from vegetation in ruralor agricultural settings. In addition, these features may also affect the conductance of thehorizontal flow term due to additional frictional resistance and small scale energy dissipationover the obstruction heights. Rill storage can be an important factor in several urban aswell as rural settings as shown in Figure 2.1c. It represents the amount of storage that mustbe filled before any lateral surface flow can occur. Microtopographic relief, relative to thescale of the finite elements in the grid, is included in rill storage and can have a substantialimpact on hydrograph shape [Woolhiser et al., 1997]. Finally, for agricultural plots or grasslands (shown in Figure 2.1d), the storage effects of rills as well as storage exclusion of thecrop must be taken into account and both rill and obstruction storage need to be includedin the model. In addition, horizontal flow term conductances may also be affected over theobstruction heights.

Rill storage and obstruction storage exclusion are both modeled by assuming that thecombined depressions and exclusions have a maximum elevation and that the area coveredby surface water varies between zero and full area, from land surface up to this maximumelevation as shown in Figure 2.2. The variation of area covered by surface water with depth(Hs) is expressed as a volumetric height which is assumed to be parabolic. The slope of thiscurve is a porosity or void ratio which varies between zero at land surface up to unity atheight Hs. Linear or other functions may have been used, however, a parabolic variationprovides for continuous derivatives at land surface and at the maximum height Hs, thusassisting during numerical solution by Newton-Raphson or modified-Picard linearizationmethods. The heights of depression storage Hd and of storage within the obstructions, Ho,may be estimated such that they geometrically represent the mean spacing (equivalent voidspace) within the respective storage elements. The depression storage height above landsurface (LS + Hd), is also used to indicate the elevation below which flow depth is zeroin the advection terms of Equation 2.45, when depression storage is modeled for a system.Thus, surface flow occurs laterally only above elevations of LS +Hd, i.e., when water levelsare above the depression storage elevations. In addition, conductance terms Kox and Koy


Figure 2.1: Treatment of Storage Terms for Various Settings.


Figure 2.2: Conceptual Model for Depression Storage and Obstruction Storage Exclusion.

may be further reduced by a factor kro above this elevation, up to the obstruction height ofstorage exclusion to account for the additional resistance losses. The factor kro varies fromzero to unity as the obstruction heights vary from 0 to Ho.

2.3 One-Dimensional Hydraulic Features

2.3.1 General

One-dimensional hydraulic features like streams, rivers, subsurface wells, water supplylines, and drain pipes are simulated by a general one-dimensional equation to describefluid flow in terms of the Hagen-Poiseuille analytic, Manning’s or Hazen-Williams empiricalformula. Common node or dual node approaches can be used to simulate the interactionbetween the 1D flow domain and 2D overland or 3D subsurface domains. The 1-D featuresin HydroGeoSphere can be applied to design urban infrastructures and regional-scaledrainage networks.



2.3.2.1 General One-Dimensional Flow

Water flow along the axis, s, of a one-dimensional features can be described by the followinggeneralized form of the continuity equation integrated over the flow area perpendicular tothe principal axis, Af :

− ∂

∂s

(Q1D

)+Qwδ

(s− sp

)= ∂

∂t

[Af]

(2.48)

and:Q1D = Af · q1D (2.49)

where Q1D is the fluid flux along the one-dimensional medium, q1D is the linear averagedvelocity over the cross sectional flow area Af and Qw is the rate of addition or extraction ofwater at s = sp . In pipe flow, the area of flow is defined as a function of saturation suchthat:

Af = A1DS1D (2.50)

where A1D is the saturated pipe area and S1D is the pipe saturation ranging between 0 and1 for empty and fully filled pipes.

Various analytical and empirical relationships have been suggested to solve one-dimensionalfluid flux for different types of media. Hagen-Poiseuille analytic formula has been used todescribe flow along the inside of wells while Manning’s equation is widely used for openchannel flow for both streams and rivers. Furthermore, the Hazen-Williams equation hasbeen typically recommended for pipe flow in water supply systems. A generalized formto describe the flow along each of the one dimensional hydraulic features is shown in thefollowing form:

Q1D = −C ·Af ·(RH

)p · [∂hw∂s

]q(2.51)

where C is a proportionality constant, p and q are fitting exponents, and RH is the hydraulicradius.

2.3.2.2 Flow Through Subsurface Wells

Groundwater wells are typically installed at the ground surface and can be used to extractgroundwater, monitor subsurface environments, or inject hazardous waste fluids for long-termstorage. Generally, the length of the subsurface wells can be assumed to be much larger thanthe cross-sectional area of the well casing. With this assumption, well hydraulic propertiesand fluid flow can be simplified, integrated, and averaged over the well cross section into aone-dimensional approximation.

Saturated groundwater flow along the axis of a well, assuming laminar conditions, has


been described using Hagen-Poiseuille analytic formula in groundwater hydrology Therrienand Sudicky [2000]:

Q1D = −(πr2

w

)(r2wρg

8µ

)∂hw∂s

(2.52)

and:∂

∂s

(πr4

w

ρg

8µ∂hw∂s

)+Qwδ(s− sp) + Γpm→w = πr2

wSsw∂hw∂t

(2.53)

where rw is the radius of a well, hw = hw(s) is the hydraulic head in the well, Ssw = ρgβ isthe specific storage coefficient of a fluid-filled borehole where ρ and β are the density andcompressibility of water, g is the gravitational acceleration, and µ is the viscosity of the fluid.For partially saturated wells, Equations 2.52 and 2.53 are modified to include a reductionof flow area, permeability, and water volume due to the decrease in water saturation asdescribed as:

Q1D = −krw(Sww · πr2

w

)(r2w

ρg

8µ

)∂hw∂s

(2.54)

and:

∂

∂s

(krwSwwπr

4w

ρg

8µ∂hw∂s

)+Qwδ(s− sp) = πr2

wSwwSsw∂hw∂t

+ πr2w

∂Sww∂t

(2.55)

where Sww is the degree of saturation and the relative permeability of the well and krwranges between 0 (when it is completely dry) and 1 (when it is fully-saturated).

Therrien and Sudicky [2000] accounted for the interaction between the well and thesurrounding porous medium by assuming and the flux continuity can also be assumed suchthat

Γpm→w = −2πrw(kr)exch(pm,w)Kexch(pm,w)hw − h

lexch(pm,w)(2.56)

and:

∂

∂s

(krwSwwπr

4w

ρg

8µ∂hw∂s

)+Qwδ(s− sp) + Γpm→w = πr2

wSwwSsw∂hw∂t

+ πr2w

∂Sww∂t

(2.57)

where Γpm→w is the fluid exchange through the interface between the porous media and wellwith an exchange the thickness of lexch(pm,w) and a hydraulic conductivity of Kexch(pm,w).The upstream relative permeability (kr)exch(pm,w) is determined between the two domainsby:

(kr)exch(pm,w) =

(kr)pm, if h ≥ hw(kr)w, if h < hw

(2.58)

2.3.2.3 Flow Through Subsurface Pipe Systems

For the design of water supply system, pressure driven flow is often described by the Hazen-Williams empirical equation (HWE). The HWE relates the flow of water and hydraulic


properties inside of the pipe to the rate of total head loss caused by friction such that

Q1D = −k · CHW ·Af · (RH)0.63 ·[∂hp∂s

]0.54(2.59)

∂

∂s

(k · CHW ·AF · (RH)0.63 ·

[∂hp∂s

]0.54)+Qpδ(s− sc) = πr2

pSsp∂hp∂t

(2.60)

where hp = hp(s) is the hydraulic head at the given location s (along the principal axis),Ssp = ρgβ is the specific storage, Qp is the rate of water consumption at the location sc,k is the dimensional unit conversion factor (0.849m0.37/s), CHW is the Hazen-Williamsroughness coefficient, and RH is the hydraulic radius defined as the ratio of the area of flowand the wetted perimeter (RH ≡ Af/Pf ). The hydraulic radius for a full-pipe flow is definedas:

RH = πr2w

2πrw= rw

2 (2.61)

Because water supply systems carry pressurized water through pipe networks, they lose acertain portion of water throughout the networks. The leakage of water can be assumed tofollow a first-order leakance relationship and is incorporated into Equation 2.60 as:

Γp→pm = −2πrpKexch(pm,p)hp − h

lexch(pm,p)(2.62)

∂

∂s

(k · CHW ·AF · (RH)0.63 ·

[∂hp∂s

]0.54)+Qpδ(s− sc) + Γp→pm = πr2

pSsp∂hp∂t

(2.63)

Note that the head in water supply pipes, hp, is supposed to be significantly greater thanthe surrounding head, h, and thus water should always leak from the water supply pipe intothe subsurface or Γp→pm.

2.3.2.4 Open Channel Flow

Open channel flow along rivers and streams in the surface or for gravity flow in sewer systemsis best described with Manning’s empirical formula:

Q1D = − 1ns·Af · (RH)2/3 ·

[∂hc∂s

]0.5(2.64)

∂

∂s

(− 1ns·AF · (RH)2/3 ·

[∂hc∂s

]0.5)= BT

∂hc∂t

(2.65)

where ns is the Manning roughness coefficient and BT is the top width of the channel water.Surface rivers and streams may interact with both the overland domain (river banks) and


also the porous medium (stream beds). Sewer systems, typically buried underground, onlyinteract directly with the porous medium domain.

Γolf→c =

0, if hc < Zbank and ho < Zbank

(4Cd/3)(hup−Zbank)√

2g(hup−hdn

sgn(ho−hc) , if hc ≥ Zbank and ho ≥ Zbank

(4Cd/3)(hup−Zbank)√

2g(hup−Zbank

sgn(ho−hc) , otherwise

(2.66)

Γpm→c = −Pf (kr)exch(pm,c)Kexch(pm,c)hc − h

lexch(pm,c)(2.67)

∂

∂s

( 1ns·AF · (RH)2/3 ·

[∂hc∂s

]0.5)+ Γpm→c + Γolf→c = BT

∂hc∂t

(2.68)

where hup and hdn are the upstream and downstream heads among h and hc , Zbank is theriver bank elevation, and Cd is the weir discharge coefficient.

2.4 Flow Coupling

Two different approaches are used to define the water exchange terms Γex between twodifferent domains. The first approach is based on superposition (see Therrien and Sudicky,1996), where continuity of hydraulic head is assumed between the two domains concerned,which corresponds to instantaneous equilibrium between the two domains. In that case, theΓex term does not need to be evaluated implicitly in the model and we do not present itsdefinition. However, the fluid exchange can be computed after the numerical solution of thediscrete equations as a post-processing step. This approach corresponds to the commonnode scheme mentioned later in the manual.

The second method is more general because it does not assume continuity of hydraulic headbetween two domains but uses a Darcy flux relation to transfer water from one domainto the other. The Darcy flux is computed from the hydraulic head difference between twodomains and assumes that they are separated by a (possibly) thin layer of porous materialacross which water exchange occurs. This second approach corresponds to the dual nodescheme mentioned later in the manual.

Table 2.1 summarizes the types of fluid flow coupling currently available in HydroGeo-Sphere. Both common node and dual node approaches are currently available in the modelto simulate the exchange between the subsurface porous medium and fractures, betweenthe porous medium and wells, and between the porous medium and the tile drains. On theother hand, fluid exchange between the subsurface porous medium and a dual continuumcan only be simulated with the dual node approach. We present here the definition of theexchange term for the dual node approach.


Table 2.1: Types of Coupling and Dimensionality for Fluid Flow.Coupling

Domains Common DualPorous medium - Discrete fractures (2-D)

√ √

Porous medium - Second continuum (3-D)√

Porous medium - Wells (1-D)√ √

Porous medium - Tile drains (1-D)√ √

Porous medium - Surface (2-D)√ √

Porous medium - Channel (1-D)√ √

2.4.1 Dual Continuum Subsurface Coupling

When the dual node approach is chosen to represent simultaneous flow in the subsurfaceporous medium and a second continuum (representing fractures or macropores), the exchangeterm can be defined as (Gerke and Van Genuchten, 1993):

Γd = αwdKakra(ψd − ψ) (2.69)

where:αwd = βd

a

[γwa

](2.70)

where βd/a is the macropore surface area per unit total volume of the medium [L−1], βd is adimensionless geometrical shape factor, a is the fracture-matrix skin thickness [L] over whichthe flow exchange occurs, and γw is a dimensionless empirical coefficient. The hydraulicconductivity of the interface between the two domains is given by Ka [L T−1] and its relativepermeability is kra. For dual-porosity systems, the geometry factor βd has been shown tobe equal to 3 for rectangular slabs and 15 for spheres. Gerke and Van Genuchten [1993]provide more detail on the evaluation of the exchange term.

2.4.2 Surface - Subsurface Coupling

When the dual node approach is chosen to represent simultaneous flow in the subsurfaceand the surface domain, the exchange term is given by:

doΓo = wmkrKzz

lexch(h− ho) + wd

kdrKdzz

lexch(hd − ho) (2.71)

where a positive Γo represents flow from the subsurface system to the surface system asdetermined by Equation 2.32, ho is the surface water head, h and hd are the subsurface porousmedium and dual medium heads, respectively, kr and kdr are the relative permeabilities forthe exchange flux, Kzz and Kdzz are the vertical saturated hydraulic conductivities of theunderlying porous and dual media, respectively, and lexch is the coupling length. The relativepermeability term is same as the subsurface relative permeability when water flows fromsubsurface to surface, while, when water flows from surface to subsurface it is determined by


the ratio of the water depth in the surface to the total obstruction height (Hs) such that

kr =S

2(1−Sexch)exch when do < Hs

1 when do > Hs(2.72)

where Sexch = do/Hs.

2.5 Flow Boundary Conditions

2.5.1 Subsurface flow

Boundary conditions for subsurface flow include the following: first-type (Dirichlet) bound-aries of prescribed hydraulic or pressure head, areal infiltration or recharge, source/sinks,evaporation, seepage faces, free-drainage and drain nodes, and third-type boundary con-ditions. The boundary conditions can also be allowed to vary in time. Details of theimplementation of these boundary conditions in the model are given in the next chapter.

When HydroGeoSphere is used to solve fully coupled subsurface and surface flow, the arealrecharge boundary for subsurface flow is not required since the solution to the interactingsystem determines the subsurface recharge.

2.5.2 Surface Flow

Boundary conditions to the surface flow system include the following: first-type (Dirichlet)boundaries of prescribed water elevation, direct rainfall inputs, source/sinks, evaporation,zero-depth gradient and nonlinear critical-depth conditions, as discussed in the next chapter.

2.5.3 Interception and Evapotranspiration

Interception and comprehensive evapotranspiration [Panday and Huyakorn, 2004] are sim-ulated as mechanistic processes governed by plant and climatic conditions as noted byKristensen and Jensen [1975], and Wigmosta et al. [1994].

2.5.3.1 Interception

Interception is the process involving retention of a certain amount of precipitation on theleaves, branches, and stems of vegetation or on buildings and structures in urban areas. Theinterception process is simulated by the bucket model, wherein precipitation in excess ofinterception storage and evaporation from interception reaches the ground surface. Theinterception storage varies between zero and SMax

int , the interception storage capacity [L],which depends on the vegetation type and its stage of development. It is calculated from[Kristensen and Jensen, 1975]

SMaxint = cintLAI (2.73)


where LAI is the dimensionless leaf area index and cint is the canopy storage parameter [L].

Note that LAI represents the cover of leaves over a unit area of the ground surface, andmay be prescribed in a time-dependent manner. The interception storage is filled by rainfalland depleted by evaporation.

2.5.3.2 Evapotranspiration

Evapotranspiration is rigorously modeled as a combination of plant transpiration and ofevaporation, and affects both surface and subsurface flow domains. Transpiration fromvegetation occurs within the root zone of the subsurface which may be above or below thewater table. The rate of transpiration (Tp) is estimated using the following relationship thatdistributes the net capacity for transpiration among various factors [Kristensen and Jensen,1975]:

Tp = f1(LAI) f2(θ)RDF [Ep − Ecan] (2.74)

where f1(LAI) is a function of leaf area index [dimensionless], f2(θ) is a function of nodalwater content [dimensionless] and RDF is the time-varying root distribution function. Thevegetation term is expressed as:

f1(LAI) = max0,min[1, (C2 + C1 LAI)] (2.75)

The root zone term is defined by the relationship:

RDF =∫ z′

2z′

1rF (z′)dz′∫ Lr

0 rF (z′)dz′(2.76)

and the moisture content dependence term is expressed as

f2(θ) =

0 for 0 ≤ θ ≤ θwpf3 for θwp ≤ θ ≤ θfc1 for θfc ≤ θ ≤ θof4 for θo ≤ θ ≤ θan0 for θan ≤ θ

(2.77)

where:

f3 = 1−[θfc − θθfc − θwp

]C3

(2.78)

f4 =[θan − θθan − θo

]C3

(2.79)

and where C1, C2, and C3 are dimensionless fitting parameters, Lr is the effective root length[L], z′ is the depth coordinate from the soil surface [L], θfc is the moisture content at fieldcapacity, θwp is the moisture content at the wilting point, θo is the moisture content atthe oxic limit, θan is moisture content at the anoxic limit and rF (z′) is the root extractionfunction [L3 T−1], which typically varies logarithmically with depth.


The function f1 correlates the transpiration (Tp) with the leaf area index (LAI) in a linearfashion. The function f2 correlates Tp with the moisture state of the roots and is an extensionof the function of Kristensen and Jensen [1975] to account for root processes in greater detail.Below the wilting-point moisture content, transpiration is zero; transpiration then increasesto a maximum at the field-capacity moisture content. This maximum is maintained up tothe oxic moisture content, beyond which the transpiration decreases to zero at the anoxicmoisture content. When available moisture is larger than the anoxic moisture content, theroots become inactive due to lack of aeration [Feddes et al., 1978]. In general, f2(θ) is anonlinear function of θ, though the ramping function is linear when C3 = 1.

In order to incorporate root growth or time-varying root depths (Lr(t)), the equation tocalculate the transpiration rate needs to be modified as

Tp = f1(LAI)f2(θ)RDF (t)[Ep − Ecan],

where the time-varying root density function (RDF (t)) is defined as

RDF (t) = Lmfr(t) = LmL0L0 + (Lm − L0)e−rt ,

where Lm is the maximum root depth, fr(t) is the root growth coefficient, L0 is the initialroot depth, and r is the classical Verhulst–Pearl logistic growth rate.

Two models are provided for evaporation. The first model assumes that evaporation occurs ifthe reference evapotranspiration Ep has not been removed by the above processes of canopyevaporation Ecan and plant transpiration Tp. Therefore, evaporation from ground surfaceEOLF and the subsurface soil layers Es is estimated as follows:

EOLF = α∗OLF (Ep − Ecan − Tp) (2.80)

Es = α∗(Ep − Ecan − Tp − EOLF )EDF (2.81)

The second model assumes that evaporation occurs along with transpiration, resulting fromenergy that penetrates the vegetation cover and is expressed as

EOLF = α∗OLF (Ep − Ecan)[1− f1(LAI)]EDF (2.82)

Es = α∗(Ep − Ecan − EOLF )[1− f1(LAI)]EDF (2.83)where α∗OLF and α∗ are the wetness factors for surface and subsurface given by

α∗OLF =

( dohs

)2(1−do/hs) for 0 ≤ do ≤ ds1 for do > ds

(2.84)

α∗ =

θ−θe2θe1−θe2

for θe2 ≤ θ ≤ θe11 for θ > θe10 for θ < θe2

(2.85)

where θe1 is the moisture content at the end of the energy-limiting stage (above which fullevaporation can occur) and θe2 is the limiting moisture content below which evaporation iszero [Allen et al., 1998].


Equation 2.85 expresses the moisture availability term for the subsurface domain. For theoverland flow domain, α∗ is calculated as varying between unity when the elevation of flowis at or above depression storage zo +HD and zero for a flow elevation at the land surface(zo), thus representing the reduced evaporative area of available water in the overland flowdomain within the depressions.

The term EDF in Equation 2.81 or 2.83 is the evaporation distribution function thatincludes the overland and subsurface flow domains. Two alternative conceptualizations areprovided for EDF . For the first model, it is assumed that the capacity for evaporation((Ep−Ecan− Tp) in Equation 2.81 or (Ep−Ecan)(1− f1(LAI)) in Equation 2.83) decreaseswith depth below the surface (subject to available moisture) due to the reduction of energypenetration in the soil. Therefore, an appropriate EDF may be prescribed as a function ofits depth from land surface. For the second model, the capacity for evaporation is met fromthe land surface downward to a prescribed extinction depth (Bsoil).

2.6 Winter Hydrological Processes

2.6.1 Surface Water Flow with Snowmelt

Depth integrated surface water flow equation describes only the mass balance of the liquidwater in the overland flow domain. In order to consider both solid and liquid phases ofwater in the surface flow domain, the governing equation needs to be expanded to includeboth water and snow mass (ρwdwv and ρsnowdsnow). If the solid phase snow is assumed tobe immobile, the mass balance of the total water can be formulated as the following:

∂ (ρwdwv + ρsnowdsnow)∂t

=∂

∂x

(ρwdfKox

∂ho∂x

)+ ∂

∂y

(ρwdfKoy

∂ho∂y

)− ρwΓex + ρwQo + ρsnow (Qsnow − µ)

(2.86)

where Qsnow and µsnow represent the rates of snow precipitation and sublimation per unitsurface area [L3 T−1 L−2]. The depth of snow is determined by the rates of snow precipitation,sublimation, and melting (always sink) which is caused by temperature change.

∂

∂t(ρsnowdsnow) = ρsnow (Qsnow − µ)− ρsnowQmelt (2.87)

where the depth of snow is always positive and the rate of melting is assumed to beproportional to a constant (η) and the difference between air temperature (Tair ) andthreshold temperature (Tthreshold ) when Tair > Tthreshold or η = 0 when Tair ≤ Tthreshold .

ρsnowQmelt = η (Tair − Tthreshold) (2.88)

By combining the total water mass balance equation with the snow balance equation, the


balance equation for the liquid phase water is derived as

∂ (ρwdwv)∂t

= ∂

∂x

(ρwdfKox

∂ho∂x

)+ ∂

∂y

(ρwdfKoy

∂ho∂y

)− ρwΓex + ρwQo + ρsnowQmelt

(2.89)

The above equation indicates that the snowmelt always acts as a source for the liquid waterbalance in the surface flow domain.

2.6.2 Groundwater Flow with Freezing and Thawing of Pore Water

In the subsurface flow system, a conventional groundwater flow equation describes the massbalance of only the liquid water in a three-dimensional porous medium. When the liquidphase of water can be transformed into the solid phase ice (i.e., freezing) or vice versa (i.e.,melting), the total mass of water in the system is ρwθsSw + ρiceθsSice where the subscriptsice represents the solid phase ice. If the ice is assumed to be immobile, then the balance ofthe total water mass can be described by the following equation:

∂

∂t(ρwθsSw + ρiceθsSice) = −∇ · ρwq + (ρwQw) (2.90)

where ρice and Sice are the density and saturation of solid phase ice in the porous medium.

The partitioning of water between solid and liquid phases is assumed to be determined onlyby the temperature (which is a function of time at a given point) such that

p(T ) = ρiceSiceρwSw + ρiceSice

= 0 if T ≥ Tf

=(Tf − TTf − Tm

)αif Tm ≤ T < Tf

= 1 if T < Tm

(2.91)

where Tf and Tm are the freezing and melting temperatures. If the density difference betweensolid and liquid phases of water is assumed to be negligible and the total water saturationremains constant at a given time at a given location,

∂

∂t(ρiceθsSice) = ∂p(T )

∂tθs [Swρw + Siceρice] (2.92)

By subtracting the balance equation of ice mass from the balance equation of the total watermass, mass balance for the liquid phase water is derived as the following:

∂

∂t(ρwθsSw) = −∇ · ρwq + (ρwQw)− ∂p(T )

∂tθs [Swρw + Siceρice] (2.93)

where the last term for the freezing or melting shows that the freezing of water (withdecreasing temperature) plays a role of sink for the liquid phase water while the meltingwith increasing temperature being a source for the liquid phase.


A simple one-dimensional analytical model is employed to determine the vertical distributionof the temperature of bulk porous medium.

∂

∂t(Tpm − Tb) = ∂

∂z

(kpmcpm

∂(Tpm − Tb)∂z

)(2.94)

where Tpm is the temperature of the bulk porous medium, kpm and cpm are the bulk thermalconductivity and heat capacity, respectively, and it is assumed that the temperature atdepth is given as Tb and the surface temperature is same as the atmospheric temperature(Tatm) such that

Tpm(z = 0, t) = Tatm(t) (2.95)Tpm(z 0, t) = Tb (2.96)

The analytical solution of the equation is given as follows:

Tpm(z, t) = Tb + z√4πκ

∫ t

τ=0

∂Tatm(τ)∂τ

erfc(

z√4κ(t− τ)

)dτ

= Tb + z√4πκ

∫ t

τ=0Tatm(τ)e

−z2/4κ(t−τ)

(t− τ)3/2 dτ

(2.97)

where the thermal diffusivity κ is defined as kpm/cpm.

2.6.3 Dual Continuum Freezing and Thawing of Pore Water

Groundwater flow and with freezing and thawing in the three-dimensional porous mediumdomain (Equation 2.90) is extended to include a dual continuum linked by a fluid exchangeterm:

wd∂

∂t(ρwθsSw,d + ρwθsSice,d) = −∇ · (wdρwqd) + ρwQd + ρwΓd (2.98)

The fluid exchange term is proportional to the head difference between the two domains, therelative hydraulic conductivity of the interface, and the ice content of the porous mediumdomain:

Γd = αwdKa(1− Sice)kr(ψd − ψ) (2.99)

Partitioning of water between solid and liquid phases in the porous medium domain isassumed to be determined only by the temperature (which is a function of time at a givenpoint), as given previously in Equation 2.91. Within the dual continuum domain, twooptions for freezing and thawing exist. The first option uses the same formulation as usedin the porous medium domain (default). The second option gives the rate of ice-formationafter Mohammed et al. [2020] and is controlled by the soil matrix temperature (T ), and aspecified thermal transfer coefficient (α∗) according to

∂θice,d∂t

= −α∗T (2.100)


The thermal transfer coefficient α∗ [T−1 °C−1] may be calculated by

α∗ = αwdλmρiceLiwd

, (2.101)

where Li (= 335,000 J kg−1) is the enthalpy of fusion for water, λm [W m−1 °C−1] isthe thermal conductivity of the domain interface, which is assumed equal to the thermalconductivity of the matrix domain, and αwd is the exchange coefficient that is defined basedon the geometry of heat transfer between domains (Equation 2.70).

2.7 Solute Transport


We present here the basic subsurface transport equation solved by HydroGeoSpherewhich is expanded in its use to incorporate, among other features, discrete fractures, asecond porous continuum, wells, tile drains, and the surface flow domain.

2.7.1.1 Porous Medium

Three-dimensional transport of solutes in a variably-saturated porous matrix is described bythe following equation:

−∇ · wm (qC − θsSwD∇C) + [wmθsSwRλC]par +∑

Ωex ±Qc

= wm

[∂(θsSwRC)

∂t+ θsSwRλC

] (2.102)

where C is the solute concentration [M L−3] of the current species amongst possibly multiplespecies and λ is a first-order decay constant [L−1]. The subscript par designates parentspecies for the case of a decay chain. For the case of a straight decay chain, there isonly one parent species, as might be the case for a radioactive decay chain; however, fordegrading organic species, a particular species may have several parent sources through acomplex degradation process. Solute exchange with the outside of the simulation domain,as specified from boundary conditions, is represented by Qc [M L−3 T−1] which representsa source (positive) or a sink (negative) to the porous medium system. The assumption offluid incompressibility is made in Equation 2.102. The dimensionless retardation factor, R,is given by [Freeze and Cherry, 1979]:

R = 1 + ρbθsSw

K ′ (2.103)

where ρb is the bulk density of the porous medium [M L−3] and K ′ is the equilibriumdistribution coefficient describing a linear Freundlich adsorption isotherm [M−1 L3]. Notethat for variably-saturated conditions, the water saturation appears in the definition of R.


The hydrodynamic dispersion tensor D [L2 T−1] is given by [Bear, 1972]:

θsSwD = (αl − αt)qq|q| + αt|q|I + θs Sw τDfree I (2.104)

where αl and αt are the longitudinal and transverse dispersivities [L], respectively, |q| is themagnitude of the Darcy flux, τ is the matrix tortuosity [-], Dfree is the free-solution diffusioncoefficient [L2 T−1] and I is the identity tensor. The product τDfree represents an effectivediffusion coefficient for the matrix. In the unsaturated zone, the tortuosity is allowed to varywith the water saturation, Sw, according to the Millington-Quirk relationship [Millingtonand Quirk, 1961], given by:

τ = (Swθs)7/3/θ2s (2.105)

In Equation 2.102, Ωex represents the mass exchange rate of solutes per unit volume [ML−3 T−1] between the subsurface domain and all other types of domains supported by themodel. Currently, these additional domains are surface, wells, tile drains, channels, discretefractures, immobile second continuum and mobile dual continuum. The definition of Ωexdepends on the conceptualization of solute exchange between the domains and will be definedlater. In the equations shown for the other domains, we will use the notation ex=f, ex=im,ex=d, ex=w, ex=t, ex=c ex=o, for the fracture, immobile continuum, dual continuum, well,tile drain, channel and surface domains, respectively.

2.7.1.2 Discrete Fractures

The equation for two-dimensional solute transport in a variably-saturated fracture followsfrom the equation describing solute transport in a fully-saturated fracture [e.g. Tang et al.,1981; Sudicky and McLaren, 1992, Therrien and Sudicky, 1996]. Its form is:

−∇ · (wfqfCf − wfSwfDf∇Cf ) + wf [RfλfCf ]par − wfΩf =

wf

[∂(SwfRfCf )

∂t+ SwfRfλfCf

](2.106)

where Cf is the concentration in a fracture [M L−3], λf is a first-order decay constant [L−1]and Df is the hydrodynamic dispersion tensor of the fracture [L2 T−1]. An expression similarto Equation 2.104 can be used to represent Df , where dispersivities and fluxes correspondto those of the fracture and the fracture porosity is assumed to be unity. The dimensionlessretardation factor Rf is defined according to [Freeze and Cherry, 1979]:

Rf = 1 +2K ′fwf

(2.107)

where K ′f is a fracture-surface distribution coefficient [L], which is defined as the ratiobetween the mass of solute on the solid phase per unit area of an assumed planar fracturesurface over the concentration of a solute in solution.


2.7.1.3 Double Porosity

The model can simulate double-porosity transport with the classical first-order theory (seeSudicky [1990]), which divides the subsurface domain into a mobile and an immobile region.The formulation used here for double-porosity is restricted to steady-state flow conditions.It is assumed that the porous medium represents the mobile domain, where flow is describedby Equation 2.1 and solute transport is described by Equation 2.102. It is also assumedthat there is no flow in the immobile domain, therefore no equation is required for immobileflow, and solute mass balance is given by:

∂(θImmCImm)∂t

− ΩImm = 0 (2.108)

where CImm is the solute concentration in the immobile region [M L−3], θImm is the porosityof the immobile region [-] and ΩImm represents the solute exchange flux between the mobileand immobile zones [M L−3 T−1].

2.7.1.4 Isotopic Fractionation

The model can simulate isotope fractionation, using a first-order kinetic formulation thatis mathematically similar to double-porosity transport. In analogy to double-porosity, weassume here that the mobile domain represents the water phase and the immobile domain isassociated with the solid, or rock, phase.

Similarly to double-porosity transport, the formulation used here for isotope fractionationis restricted to steady-state flow conditions. Fractionation from the water phase is alsorestricted to a single solid (or rock) phase. It is assumed that the porous medium representsthe water domain, where flow is described by equation (2.1) and isotope transport is describedby equation (2.102). It is also assumed that there is no isotope flow in the solid phase,therefore no equation is required for flow in the solid phase.

Mass balance for the isotope in the solid phase is given by:

∂CImm∂t

− ΩImmxr

= 0 (2.109)

where CImm is the isotope concentration in the solid, or immobile region [M L−3], ΩImmrepresents here the isotope exchange flux between the mobile (water) and immobile (solid)zones [M L−3 T−1], and xr is the dimensionless mass ratio of the isotope in the solid phaseto that in the water phase, for a unit volume of water-saturated rock of constant porosity.

2.7.1.5 Dual Continuum

When the dual continuum option is used, advective-dispersive solute transport can besimulated in a second continuum based on the formulation presented by Gerke and VanGenuchten [1993]. Note that, as opposed to the double-porosity option, the dual continuum


option is not restricted to steady-state flow conditions because it allows transient fluid ex-change between the two interacting continua. Also, fluid flow and advective-dispersive solutetransport can be simultaneously solved in the porous medium and the second continuum.Three-dimensional transport of solutes in a variably-saturated dual continuum is describedby:

−∇ · wd (qdCd − θsdSwdDd∇Cd) + [RdλdCd]par − wdΩd ±Qcd =

wd

[∂(θsdSwdRdCd)

∂t+ θsdSwdRdλdCd

](2.110)

where Cd is the solute concentration in the dual continuum [M L−3] and λd is a first-orderdecay constant [L−1]. Solute exchange with the outside of the simulation domain, as specifiedfrom boundary conditions, is represented by Qcd [M L−3 T−1] which represents a source(positive) or a sink (negative) to the dual continuum. The dimensionless retardation factor,Rd, is given by [Freeze and Cherry, 1979]:

Rd = 1 + ρbdθsdSwd

K ′d (2.111)

where ρbd is the bulk density of the dual continuum [M L−3] and K ′d is the equilibriumdistribution coefficient describing a linear Freundlich adsorption isotherm [M−1 L3]. Notethat for variably-saturated conditions, the water saturation appears in the definition of Rd.

The hydrodynamic dispersion tensor Dd [L2 T−1] is given by [Bear, 1972]:

θsdSwdDd = (αld − αtd)qdqd|qd|

+ αtd|qd|I + θsd Swd τdDfree I (2.112)

where αld and αtd are the longitudinal and transverse dispersivities [L], respectively, |qd|is the magnitude of the Darcy flux, τd is the dual continuum tortuosity [-], Dfree is thefree-solution diffusion coefficient [L2 T−1] and I is the identity tensor. The product τdDfree

represents an effective diffusion coefficient for the dual continuum. Recall from the previousdiscussion on flow in the unsaturated zone that the tortuosity is a function of the degree ofsaturation according to Equation 2.105.

2.7.1.6 Wells

One-dimensional solute transport along the axis of a well is described by:

−∇ · πr2s

(qwCw − SwwDw∇Cw

)+ πr2

s [λCw]par −

Qw(Cw − CwInj)δ(l − l′)− πr2

sΩw = πr2s

∂Cw∂t

+ πr2sλCw (2.113)

where Cw is the solute concentration in the well [M L−3], qw is the fluid flux along the wellaxis [L T−1], and CwInj is the concentration of injected water.

The dispersion coefficient for the well, Dw [L2 T−1] is equal to [Lacombe et al., 1995]:

Dw = r2sqw

2

48Dfree+Dfree (2.114)


2.7.1.7 Tile Drains

One-dimensional solute transport along the axis of a tile drain is described by:

−∇ ·A(qtCt − SwtDt∇Ct

)+A [λCt]par −AQt(Ct−CtInj)δ(l− l

′)−AΩt = ∂ACt∂t

+AλCt

(2.115)where Ct is the solute concentration in the tile drain [M L−3], qt is the fluid flux along thedrain axis [L T−1], and CtInj is the concentration of injected water.

The dispersion coefficient for the drain, Dt [L2 T−1] has a form similar to that for aone-dimensional well and is equal to:

Dt = r2tqt

2


where rt is an equivalent radius for a tile drain of cross-section A.

2.7.1.8 Surface runoff

The equation for two-dimensional transport of solutes along the surface domain is written as

−∇(qoCo−Doφoho∇Co) + [φohoRoλCo]par−doΩo = ∂

∂t(φohoRoCo) +φohoRoλCo (2.117)

where Co is the concentration in water on the surface domain [M L−3], Do is the hydrodynamicdispersion tensor of the surface flow domain [L2 T−1] and ∇ is the vertically integratedtwo-dimensional gradient operator. An expression similar to Equation 2.104 is used torepresent the dispersion coefficient Do and the retardation factor Ro is represented by anexpression similar to Equation 2.107.

2.7.1.9 Channels

One-dimensional solute transport along the axis of a channel is described by:

−∇·A(qcCc − SwcDc∇Cc

)+A [λCc]par−AQc(Cc−CcInj)δ(l− l

′)−AΩc = ∂ACc∂t

(2.118)

where Cc is the solute concentration in the channel [M L−3], qc is the fluid flux along thechannel axis [L T−1], and CcInj is the concentration of injected water.

The dispersion coefficient for the channel, Dc [L2 T−1] has a form similar to that for aone-dimensional well and is equal to:

Dc = r2cqc

2


where rc is an equivalent radius for a channel of cross-section A.


2.7.1.10 Thermal Energy Transport

Recent research has highlighted the importance of quantifying the impacts of climate changeon the hydrological and ecological regimes, prompting the need for a model capable ofassessing thermal energy transport in a holistic manner. The fully-integrated frameworkof HydroGeoSphere made it an ideal choice for integrating the thermal energy transportsystem. HydroGeoSphere is a fully-integrated three-dimensional surface/subsurface flowand transport model. It is capable of two-dimensional overland/stream flow and transportand three-dimensional variably-saturated flow and transport in the subsurface includingmacropores, fractures and karst conduits. These media are fully-coupled and employ asimultaneous solution of the surface/subsurface flow and transport equations using a Control-Volume Finite Element method. Solving the surface and subsurface equations simultaneouslyfor both flow and transport allows for a complete coupling of the interactions within andbetween the domains. These interactions can play a very important role in assessing theimpacts of hydrological, chemical or thermal stressors. Without the integrated approach,these interactions cannot be properly quantified because some form of abstraction of thephysical processes would be necessary.

Graf [2005] incorporated heat transport within the saturated-zone flow regime into Hydro-GeoSphere together with temperature-dependent fluid properties, such as viscosity anddensity. The model’s capability was successfully demonstrated for the case of thermoha-line flow and transport in porous and fractured porous media [Graf and Therrien, 2007].HydroGeoSphere has been further modified to include thermal energy transport in theunsaturated zone and in the surface water, which is considered a key step in the linkagebetween the atmospheric and hydrologic systems. Surface heat fluxes from atmosphericinputs are an important source/sink of thermal energy, especially to the surface water system.Surface heat fluxes across the land surface were also incorporated into HydroGeoSphere.The following sections will give a brief overview of the equations used to simulate the thermalenergy transport in HydroGeoSphere.

The equation describing thermal energy transport in the unsaturated zone is similar tothat for the saturated zone, with the inclusion of a saturation term in the bulk transportparameters. The general equation for variably-saturated subsurface thermal energy transportfollowing Molson et al. [1992] is given by:

(∂ρbcbT )∂t

= −∇[qρwcwT − (kb + cbρbD)∇T ]±QT + Ωo (2.120)

where ρ is the density, c is the heat capacity, T is the temperature of the bulk subsurface, tis time, q is the Darcy flux in the subsurface, k is the thermal conductivity term, D is thethermal dispersion term, QT is a thermal source/sink and Ωo is the thermal surface/subsurfaceinteraction term, which will be discussed in a following section. The subscript b denotesa bulk term, whereas w represents the aqueous phase. In equation 2.120 1 the Darcy flux(q) in the subsurface is calculated from a numerical solution of the mixed form of Richards’equation, which is valid both above and below the water table, and the water saturationterm is included in each of the bulk property terms. The thermal dispersion term (D) iscalculated for each flow direction, and for cross-terms to account for anisotropic systems,


following [Burnett and Frind, 1987]:

Dxx = αl(q2x)/|q|+ αth(q2

y)/|q|+ αtv(q2z)/|q|+D∗ (2.121a)

Dyy = αth(q2x)/|q|+ αl(q2

y)/|q|+ αtv(q2z)/|q|+D∗ (2.121b)

Dzz = αtv(q2x)/|q|+ αtv(q2

y)/|q|+ αl(q2z)/|q|+D∗ (2.121c)

Dxy = Dyx = (αl − αth)(qxqy)/|q| (2.121d)

Dxz = Dzx = (αl − αtv)(qxqz)/|q| (2.121e)

Dyz = Dzy = (αl − αtv)(qyqz)/|q| (2.121f)

where αl, αth, and αtv are the longitudinal, horizontal transverse and vertical transversedispersivities, |q| is the magnitude of the Darcy flux, and D∗ is the effective moleculardiffusion coefficient. The bulk parameters, with the exception of thermal conductivity, arecalculated by a volumetric average approximation; for example:

ρb = (1− θ)ρs + Swθρw + (1− Sw)θρa (2.122)

where θ is porosity, Sw represents the water saturation and the subscripts s and a representthe matrix solids and the air phase, respectively.

The bulk thermal conductivity term (kb) can either be specified, or calculated. It representsthe thermal conductivity of the entire cell, which can include the matrix solids, the aqueousphase and air. Specifying a bulk thermal conductivity may be appropriate for saturatedflow conditions, when the phase composition of the subsurface does not vary with space ortime; however under variably-saturated flow conditions, the saturations of air and waterin the subsurface may change with time, and thus the thermal conductivity may alsochange. Calculating a three-phase thermal conductivity is not straightforward; severalapproaches have been presented, and no one method has proven to consistently provide morerepresentative solutions than others [Chaudhary and Bhandari, 1968, Markle et al., 2006].

Three methods of calculating the bulk thermal conductivity are available in HydroGeo-Sphere, allowing the user to determine which method is appropriate for their simulations.The first method, which is the default, uses a volumetric average approximation similar tothat employed in SUTRA [Voss, 1984]:

kb = (1− θ)ks + Swθkw + (1− Sw)θka (2.123)

The second approximation extends the accepted two-phase thermal conductivity calculationprovided by Sass et al., [1971] by calculating both a dry and saturated thermal conductivity(kdry and ksat, respectively), and then determining the bulk thermal conductivity using ageometric interpolation between the dry and saturated thermal conductivities based on thedegree of water saturation:

kdry = k(1−θ)s kθa (2.124)

ksat = k(1−θ)s kθw (2.125)

kb = Swksat + (1− Sw)kdry (2.126)


The third method uses the following non linear relationship developed by Cote and Konrad[2005]:

kb = kdry (ksat − kdry) kbr (2.127)

where kbr is a relative thermal conductivity given by

kbr = kCSw1 + (kC − 1)Sw

(2.128)

and where kC is a fitting parameter. Typical values of kC are 3.55 for a medium to finesand, and 1.9 for silty and clayey soils [Cote and Konrad, 2005]

The equation describing thermal energy transport in the surface water is similar to that forsolute transport in the surface water and is given by:

∂ρwcwdoTo∂t

= −∇[qoρwcwTo − (kw +Doρwcw)do∇To] + Eatm ±QTo − doΩo (2.129)

where d is the depth of the surface water, Eatm represents the atmospheric inputs to thesurface thermal energy system, and the subscript o denotes overland (or stream) flow.Both solute and thermal energy transport are depth averaged in the surface domain withinHydroGeoSphere. In equation 2.129, the overland water flux (qo) is calculated from thenumerical solution of the Diffusion-wave equation, together with Manning’s equation, whichis fully coupled to the Richards’ equation describing subsurface flow. These derivations forthermal energy transport are hydrodynamically, and not thermodynamically based, and arenot applicable to high temperature, high pressure hydrothermal conditions, but are valid formost shallow groundwater/surface water systems. In addition, the surface thermal regime isdepth-averaged and cannot represent thermal stratification in surface water bodies.

Atmospheric Inputs

The inclusion of atmospheric thermal energy inputs is necessary to properly simulate thesurface and subsurface temperature regimes. In this work, the atmospheric inputs fromCLASS [Verseghy, 1991] are used to determine the surface heat fluxes in HydroGeoSphere.CLASS is a well established land surface scheme, and the atmospheric inputs incorporatedinto CLASS have been demonstrated to be representative [Verseghy, 1991]. In addition tothe reliability of the CLASS approach, the equations used for atmospheric inputs are alsocomputationally inexpensive, relative to other atmospheric models, and thus are appropriatefor implementing into HydroGeoSphere. The total atmospheric input to the surfacethermal energy system can be expressed as:

Eatm = K∗ + L∗ +QH +QE (2.130)

where K∗ is the net shortwave radiation, L∗ is the net longwave radiation, QH is the sensibleheat flux and QE is the latent heat flux . All of the equations used here are taken fromCLASS [Verseghy, 1991] with the exception of the incoming longwave radiation calculation,which is adapted from Fassnacht et al. [2001]. All of the atmospheric thermal inputs arecalculated explicitly, and are treated as a source/sink term in the thermal energy transport


equation for the surface regime. A limitation to this approach is that the model does notprovide feedback to the atmosphere; however, given the scale of the atmospheric regime, itis assumed that the feedback from the smaller hydrologic domains is negligible.

Shortwave radiation is the radiant energy in the visible, near-ultraviolet and near-infraredwavelengths from the atmosphere to the Earth’s surface, broadly defined as between 0.1 and5.0 micrometers. The equation used to calculate the thermal energy input to the surfacefrom net shortwave radiation follows Verseghy [1991] and is:

K∗ = (1− αg)K↓ (2.131)

where αg is the ground surface albedo, and K↓ is the incoming shortwave radiation. Groundsurface albedo is dependent on the water content of the surficial soil, as formulated by Idsoet al.[1975], and is given by:

αg = Sw(1−θ)(αsat−αdry)0.20 + αdry for Sw(1− θ) < 0.20

αg = αsat for Sw(1− θ) ≥ 0.20(2.132)

where αsat and αdry are the limiting wet and dry soil albedoes.

A variation of this equation is also available in HydroGeoSphere to account for cloud andcanopy cover (given by Cc):

K∗ = (1− Cc)(1− αg)K↓ (2.133)

The cloud and canopy cover term (Cc) varies between 0 and 1 and represents the fractionof the sky that is blocked by either clouds or vegetation from the ground surface. NetLongwave Radiation (L∗) Longwave radiation is the infrared energy emitted by the earthand atmosphere at wavelengths between about 5 and 25 micrometers. The equation for netlongwave radiation to the surface following Verseghy [1992] is:

L∗ = L↓ − σT 4g (2.134)

where L↓ is the incoming longwave radiation, σ is the Steffan-Boltzmann constant, and Tg isthe ground surface temperature (temperature of the surface regime). In the original CLASSformulation, the incoming radiation term is specified as an input parameter. However,due to the lack of longwave radiation data (incoming longwave radiation is not routinelymeasured), incoming longwave radiation can also be calculated in HydroGeoSphere usingthe formulation given by Fassnacht et al. [2001]:

L↓ = εatσT4a (2.135)

where Ta is the air temperature, εat is the integrated emissivity of the atmosphere andcanopy, calculated by:

εat = (0.53 + 0.2055e0.5a )(1 + 0.40Cc) (2.136)

where ea is the near surface vapour pressure. Emissivity is limited so that it cannot exceed1.0, ensuring that the incoming longwave radiation is not overestimated.


Sensible heat flux represents the movement of energy from the Earth to the air abovetypically via conduction. The equation used for sensible heat flux follows Verseghy [1991]and is given by:

QH = ρacaVacD[Ta − Tg] (2.137)

where ρa is the density of the air, ca is the specific heat of the air, Va is the wind speed andcD is the drag coefficient.

Latent heat flux represents the energy transfer during the evaporation/condensation process.There are three methods of accounting for evaporation in the flow solution. The quantityof water evaporated/condensed in the flow solution provides the basis for the amount ofenergy transferred between the atmospheric and hydrologic regimes for latent heat flux.As such, the evaporation rate used in the flow solution must be linked to the latent heatflux term used to calculate the atmospheric thermal energy inputs. The first, and mostsimplistic method of accounting for evaporation, is to simply reduce the precipitation rateapplied to the surface of the domain by the evaporation rate. By not explicitly specifyingor calculating the evaporation rate in HydroGeoSphere, the latent heat flux must bedetermined independently. In this case the equation used by CLASS [Verseghy, 1991] isused, given by:

QE = LV ρaVacD[SHa − SHg] (2.138)

Where LV is the latent heat of vaporization, SHa is the specific humidity of the air, andSHg is the specific humidity of the ground surface. Specific humidity of the ground surfaceis calculated using the same formulation given by Verseghy (1991):

SHg = hSHsat[Tg] (2.139)

Where h is the relative humidity of the air in the surface soils, calculated by:

h = exp

[−gψgRwTg

](2.140)

and SHsat[Tg] is the saturation specific humidity at Tg, given by:

SHsat[Tg] = 0.622esat[Tg]pa − 0.378esat[Tg]

(2.141)

In these equations, g represents the acceleration due to gravity, ψg is the soil-water suctionat the surface, Rw is the gas constant for water vapour, esat[Tg] is the saturation vapourpressure at the ground surface and pa is the air pressure. Evaporation can also be specifiedor calculated in HydroGeoSphere. When evaporation is specified, the evaporation rate isinput to HydroGeoSphere and is subtracted from the incoming precipitation throughoutthe simulation. Internally calculating the evaporation rate is a more complex approach, basedon the empirical Hargreaves equation for determining the potential evapotranspiration, andthen calculating the actual evapotranspiration as a combination of plant transpiration andevaporation from the surface and the subsurface domains. The equations for this approachare given in detail by Li et al., (2008). Whether the evaporation is specified or calculated,the latent heat flux is then determined from the evaporation rate used in the flow solution,


Table 2.2: Types of Coupling and Dimensionality Solute Transport.Coupling

Domains Common DualPorous medium - Discrete fractures (2-D)

√ √

Porous medium - Second continuum (3-D)√

Porous medium - Double porosity (3-D)√

Porous medium - Wells (1-D)√ √

Porous medium - Tile drains (1-D)√ √

Porous medium - Surface (2-D)√ √

Porous medium - Channel (1-D)√ √

ensuring continuity between the flow and thermal transport simulations. The latent heatflux equation for a specified or calculated evaporation rates is:

QE = LVEfluxρw (2.142)

where Eflux is the specified or calculated evaporation rate.

2.8 Solute Transport Coupling

Similarly to fluid flow, two different approaches are used to define the solute exchange termsΩex between two different domains. The first approach is based on a numerical superpositionprinciple (see Therrien and Sudicky, 1996), where continuity of solute concentration isassumed between the two domains concerned, which corresponds to instantaneous equilibriumbetween the two domains. In that case, the Ωex term does not need to be evaluated explicitlyin the model and we do not present its definition. However, the solute exchange flux betweendomains can be computed after the numerical solution at a given time step. This approachcorresponds to the common node scheme.

The second method is more general because it does not assume continuity of concentrationbetween two domains, but uses a first-order expression to approximate Fickian transportto transfer solute from one domain to the other. This second approach corresponds to thedual-node scheme mentioned later in the manual.

Table 2.2 summarizes the types of solute transport coupling currently available in Hydro-GeoSphere. Solute exchange between the subsurface porous medium and the immobileregion (double-porosity option), and between the subsurface porous medium and a dualcontinuum is only simulated by the dual-node approach. For solute exchange between thedomains, both the common-node approach as well as the dual-node approach are availableoptions for coupling. We present here the definition of the exchange term for the dual-nodeapproach.


2.8.1 Mobile - Immobile Region Coupling

Solute exchange between the mobile and immobile region of a porous medium (double-porosity approach) is given by:

ΩImm = αImm(C − CImm) (2.143)

where αImm is a first-order mass transfer coefficient between the mobile and immobile regions[T−1].

For fractured porous media, Sudicky [1990] presents relationships for the mass transfercoefficient as a function of fracture geometry. For example, if a porous medium is highlyfractured and the shape of the porous medium block delineated by the fracture network canbe approximated as spheres, the mass transfer coefficient can be approximated by:

αImm = 15θImmD∗Imm

r20

(2.144)

where D∗Imm is the effective diffusion coefficient in the immobile region and r0 is the radiusof a representative sphere [L]. Another expression can be given for the case of a system ofparallel fractures, with a uniform fracture spacing equal to Bf and where the porous matrixblocks are prismatic slabs:

αImm = 3θImmD∗Imm

(Bf/2)2 (2.145)

2.8.2 Dual Continuum Subsurface Coupling

When the dual-node approach is chosen to represent simultaneous transport in the subsurfaceporous medium and a dual continuum (representing fractures), the solute exchange termcan be defined as (Gerke and Van Genuchten, 1993):

Ωd = −umC − udCd (2.146)

where:um = d∗Γdφ− αswmθsSw (2.147)

and:ud = (1− d∗)Γdφ∗ + αswmθsSw (2.148)

In Equation 2.147 and 2.148, αs is a mass transfer coefficient [T−1] resembling that definedfor the double porosity option. We further define the following variables:

d∗ = 0.5(

1− Γd|Γd|

)(2.149)

and

φ = wmθsSwθtot

; φ∗ = (1− wm)θsdSwdθtot

; θtot = wmθsSw + (1− wm)θsdSwd (2.150)


For the case where the hydraulic heads of the porous medium and the second continuum areequivalent, then, according to Equation 2.69, the term Γd above becomes equal to 0 andthe solute exchange given by term Ωd reduces to a diffusion-type exchange between the twodomains, similar to that for the double porosity medium (see Equation 2.143).

2.8.3 Thermal Energy Coupling

The coupling of the surface and subsurface thermal continua is similar to that used foradvective-dispersive contaminant transport in HydroGeoSphere. There are two methodsof coupling the surface and subsurface continua, the common node and the dual nodeapproaches. The common node approach is based on the assumption of continuity oftemperature at the surface/subsurface interface. The dual node approach on the other handuses a first-order flux relation to transfer heat from one domain to the other. The equationfor the dual-node coupling of the surface and subsurface thermal equations follows Therrienet al. (2007) and is given by:

Ωo = ρwcwTupsΓo + αoρdwncdwn(T − To) (2.151)

where Γo represents the aqueous exchange flux between the surface and subsurface (theamount of water flowing between the two regimes) and αo is an energy transfer coefficientdetermined by the thermal dispersivity over the depth of the surface/subsurface exchangezone. The subscript ups represents the ”upstream” direction, and the subscript dwnrepresents the ”downstream” direction (i.e. Tups = T, ρdwncdwn = ρwcw when Γo > 0 andTups = To, ρdwncdwn = ρbcb when Γo < 0). The most significant difference between thesolute and the thermal transport coupling equations is the treatment of mass/energy transferbetween the surface and subsurface. The downstream parameters are used to differentiatebetween the amount of thermal energy required to change the temperature of the surface andthe subsurface domains, respectively. When the diffusive gradient is transferring thermalenergy to the subsurface, the bulk heat capacity and density parameters regulate how muchthe temperature of the bulk subsurface changes given the amount of thermal energy added.Conversely, when the diffusive gradient is transporting thermal energy to the surface, theaqueous heat capacity and density terms regulate how much the surface water temperatureincreases given the thermal energy inputs. As the bulk and aqueous heat capacity anddensity terms can be significantly different, the amount of energy required to change thetemperature of the surface and subsurface regimes can also be different. For this formulation,the downstream location (where the energy is diffusing to) determines how much thermalenergy is required to reduce the thermal gradient between regimes. The movement of asolute between the two regimes is different from heat because a non-sorbing solute has atendency to remain in the aqueous phase, whereas thermal energy tends to preferentiallytransfer into the solid phase of the porous medium, thus affecting the bulk temperature.


2.8.4 Isotopic Fractionation Coupling

Isotopic exchange between the mobile (water) and immobile (solid) region of a porousmedium is similar to the double-porosity approach and is given by:

ΩImm = xrkr(αrC − CImm) (2.152)

where kr is a reverse fractionation rate [L−1] and krαr is the forward reaction rate with αrbeing the isotope fractionation factor between water and the solid phase. ConcentrationCImm describes here the isotopic concentration in the solid phase.

2.9 Solute Transport Boundary Conditions

2.9.1 Subsurface

Boundary conditions for subsurface transport include the following: first-type (Dirichlet)boundaries of prescribed concentration, prescribed mass flux or third-type (Cauchy) bound-ary conditions. For chain-decay of solutes, Bateman’s equation is used to prescribe theconcentrations for the daughter products. The boundary conditions can also be allowed tovary in time. Details of the implementation of these boundary conditions in the model aregiven in the next chapter.

2.9.2 Surface

Boundary conditions to the surface flow system include at the moment first-type (Dirichlet)boundaries of prescribed concentration.

2.10 Travel Time Probability

2.10.1 Definitions

Groundwater age is usually defined as a relative quantity with respect to a starting locationwhere age is assumed to be zero. For a given spatial position in the reservoir, the age (A)relates to the time elapsed since the water particles entered the system at the recharge limits,where age is zero. For the same spatial position, the life expectancy (E) is defined as thetime required for the water particles to reach an outlet limit of the system. Life expectancyis therefore zero at an outlet. The total transit time (T) finally refers to the total timerequired by the same water particles to migrate from an inlet zone (T = E) to an outlet zone(T = A). The three variables A, E and T are random variables, characterized by probabilitydensity functions (PDFs) gU=A,E,T , that can be regarded as the statistical occurrence ofwater particles with respect to time, which could be observed in a groundwater sample if


any analytical procedure would allow such measurements.The travel time probability gt(t,x | t0,xi) characterizes the probability density for theamount of time (t is a random variable) required by the water particles to travel from agiven position xi (at time t = t0) to the position x. The location probability gx(x, t | xi, t0)characterizes the probability density of finding water particles at the position x (x is arandom variable) at a given time t after their release at the position xi. If the input zonecorresponds to the entire inlet Γ−, then travel time corresponds to age A. Similarly, ifthe input zone corresponds to the entire outlet Γ+, then travel time corresponds to lifeexpectancy E.The age (and/or forward travel time) and life expectancy (and/or backward travel time)PDFs can be obtained as solutions of advection-dispersion equations (ADE), by making useof specific boundary conditions.

2.10.2 Basic equations

2.10.2.1 Forward model

The age PDF at a position x in an aquifer Ω can be evaluated by solving the ADE when aunit pulse of conservative tracer is uniformly applied on the recharge area Γ− (see Fig. 2.3).The resulting breakthrough curve is the probabilistic age distribution [Danckwerts, 1953;Jury, 1990]. The pre-solution of a velocity field is performed by solving the groundwaterflow equation. The age PDF is then obtained by solving the following forward boundaryvalue problem:

∂θg

∂t= −∇ · qg +∇ · θD∇g + qIδ(t)− qOg in Ω (2.153a)

g(x, 0) = g(x,∞) = 0 in Ω (2.153b)

J(x, t) · n = (q · n)δ(t) on Γ− (2.153c)

J(x, t) · n = 0 on Γ0 (2.153d)

where g(x, t) = gA(x, t) denotes the transported age PDF [T−1], q is the water flux vector[LT−1], qI and qO are fluid source and sink terms, respectively [T−1], J(x, t) is the total agemass flux vector [LT−2], D is the tensor of macro-dispersion [L2T−1], x = (x, y, z) is thevector of Cartesian coordinates [L], t is time [T], θ = θ(x) is porosity or mobile water content[−], n is a normal outward unit vector, and δ(t) is the time-Dirac delta function [T−1],which ensures a pure impulse on Γ−. The total age mass flux vector J(x, t) is classicallydefined by the sum of the convective and dispersive fluxes:

J(x, t) = qg(x, t)−D∇g(x, t) (2.154)

The third-type (Cauchy) boundary condition (2.153c) is the most meaningful condition tosimulate the age problem since it prevents backward losses by dispersion (homogeneity ofthe condition at t = 0+).


2.10.2.2 Backward model

The life expectancy PDF satisfies the adjoint backward model of Eq. (2.153a):∂θg

∂t= ∇ · qg +∇ · θD∇g − qIg in Ω (2.155a)

g(x, 0) = g(x,∞) = 0 in Ω (2.155b)J(x, t) · n = −(q · n)δ(t) on Γ+ (2.155c)−D∇g(x, t) · n = 0 on Γ0 (2.155d)

where g(x, t) = gE(x, t) denotes the transported life expectancy PDF, and where the totallife expectancy mass flux vector J(x, t) is

J(x, t) = −qg(x, t)−D∇g(x, t) (2.156)

Eq. (2.155a) is the formal adjoint of Eq. (2.153a) [Garabedian, 1964; Arnold, 1974], knownas the ”backward-in-time” equation [Uffink, 1989; Wilson, 1997] or the backward Kol-mogorov equation [Kolmogorov, 1931]. Given the forward equation, the backward equationis technically obtained by reversing the sign of the flow field, and by adapting the boundaryconditions [Neupauer, 1999; Neupauer, 2001]. On the impermeable boundary Γ0, a third-typecondition (Cauchy) in the forward equation becomes a second-type condition (Neumann) inthe backward equation, and vice-versa. A second-type condition in the forward model willalso become a third-type condition in the backward model [Gardiner, 1983, p. 146]. Theadvection term is known to be not self-adjoint (it should be written in the form q · ∇g inEq. (2.155a)) unless flow is non-divergent. However, the backward equation can still handledivergent flow fields by means of the important sink term −qIg appearing in Eq. (2.155a).This sink term has been derived in Cornaton [2003] from the vertical averaging process ofthe general 3D backward ADE, and is consistent with the analysis of Neupauer [2001; 2002]and Wilson [1997]. Recharge by internal sources (3D or 2D vertical) or by areal fluxes (fluidsource for 2D horizontal) is introduced by the first-order decay type term −qIg, which is aconsequence of the reversed flow field. Internal sources produce a sink of life expectancyprobability, while internal sinks (term qOg in Eq. (2.153a)) do not appear in the backwardmodel since a fluid sink may not influence the life expectancy PDF.The life expectancy to a specific outlet Γn ⊂ Γ+ can be calculated by modifying the boundaryvalue problem (2.155) in such a way that the function g will characterize the probabilitydensity for the water particles to reach Γn, exclusively. This can be done by assuming amaximum intensity of probability of exit at Γn (J(x, t) · n = −(q · n)δ(t)) and a minimumintensity of probability of exit at each other outlet (J(x, t) · n = 0).

2.10.2.3 Total transit time

Since T = A+E, and since A and E are independent variables, the total transit time PDFgT is obtained by the following convolution product:

gT (x, t) =∫ t

0gA(x, τ)gE(x, t− τ)dτ (2.157)


(a) (b)

+q −qΓ+

Γ− Γ

−

Γ+

T = A + E

qI

qI

qI

qI

qI

qI

A

T = A + E

A

EE

Γ−Γ

−

Γ0 Γ

0

Figure 2.3: Schematic illustration of a groundwater reservoir Ω, with inlet (Γ−) and outlet(Γ+) boundaries: (a) Age problem with normal flow field; (b) Life expectancy problem withreversed flow field. The cross stands for a small water sample, to illustrate the randomvariable total transit time (T) as the sum of the two random variables age (A) and lifeexpectancy (E).

The field of gT characterizes the evolution of groundwater particles throughout the aquiferdomain by specifying the amount of time from recharge to discharge. At a given positionin the reservoir, the temporal evolution of the groundwater particles can be characterizedby the three PDF’s gA, gE and gT . Each function contains specific information on a timeof residence, the nature of which is a function of the spatial references that are chosen forevaluation. For instance, gA is conditioned by the inlet limit Γ−, where the variable A is nil,while gE is conditioned by the outlet limit Γ+, where the variable E is nil. For the variableT, the PDF gT is conditioned by the fact that T = A at outlet, and that T = E at inlet.

2.10.2.4 Outlet/Inlet Transit Time PDF

The representative transit time distribution ϕ(t) of the reservoir outlet zone can be defined asa flux averaged concentration [Rubin, 2003], i.e. ϕ(t) is evaluated as the flow rate-normalizedsum on Γ+ of the total age mass flux response function J resulting from a unit flux impulseon Γ−:

ϕ(t) = 1F0

∫Γ+

J · n dΓ (2.158)

where F0 denotes the total flow rate through Γ+. Similarly, the inlet transit time PDF ϕ(t)is derived by enforcing Eq. (2.158) on the inlet boundary Γ−, given that the function Jrepresents the life expectancy mass flux.


2.10.2.5 Travel Time Probabilities

The forward and backward travel time probabilities are calculated as solutions of the followingforward and backward initial value problems:

∂θg

∂t= −∇ · qg +∇ · θD∇g − qOg in Ω (2.159a)

g(x, 0) = δ(x− xi)θ

in Ω (2.159b)

[qg(x, t)−D∇g(x, t)] · n = 0 on Γ− ∪ Γ0 (2.159c)

for the forward problem, and

∂θg

∂t= ∇ · qg +∇ · θD∇g − qIg in Ω (2.160a)

g(x, 0) = δ(x− xi)θ

in Ω (2.160b)

[−qg(x, t)−D∇g(x, t)] · n = 0 on Γ+ (2.160c)

−D∇g(x, t) · n = 0 on Γ0 (2.160d)

for the backward problem. The variable xi denotes the location for the release of a unitmass.The forward and backward location probabilities [L−3] are then defined by:

gx(x, t) = θ(x)g(x, t) (2.161)

The forward and backward travel time probabilities [T−1] are defined by:

gt(t,x) = ‖q(x, t)‖A(x)g(t,x) (2.162)

where A(x) denotes the area of a control plane orthogonal to velocity, through which thetravel time probability is evaluated.

2.10.2.6 Age, Life Expectancy and Travel Time Statistics

Given an age/life expectancy/travel time PDF solution gt(t,x), the following descriptivestatistics can be post-processed (see Fig. 2.4)):

1. Mean age/life expectancy/travel time µ:

µ(x) =∫ +∞

0tgt(t,x)dt (2.163)


Figure 2.4: Descriptive statistics on the travel time PDF.

2. Standard deviation σ:

σ(x) =√∫ +∞

0t2gt(t,x)dt− µ2 (2.164)

3. Mode M(x).

2.10.2.7 Mean Age and Mean Life Expectancy Direct Solutions

Temporal moment equations can be derived from Eqs. (2.153) and (2.155). For instance, themean age equation is obtained by taking the first moment form of Eq. (2.153):

−∇ · q〈A〉+∇ · θD∇〈A〉 − qO〈A〉+ θ = 0 in Ω (2.165)

where 〈A〉 denotes mean age. It requires the following boundary conditions:

〈A〉(x) = 0 on Γ− (2.166a)

J(x) · n = 0 on Γ0 (2.166b)

The mean life expectancy equation is similarly obtained by taking the first moment form ofEq. (2.155):

∇ · q〈E〉+∇ · θD∇〈E〉 − qI〈E〉+ θ = 0 in Ω (2.167)

where 〈E〉 denotes mean life expectancy. It requires the following boundary conditions:

〈E〉(x) = 0 on Γ+ (2.168a)


−D∇〈E〉(x) · n = 0 on Γ0 (2.168b)

Mean age and mean life expectancy are continuously generated during groundwater flow,since porosity θ = θ(x) acts as a source term in Eqs. (2.165) and (2.167). This sourceterm indicates that groundwater is aging one unit per unit time, in average. Finally, meantotal transit time (from inlet to outlet) can be obtained by taking the first moment form ofEq. (2.157), yielding to 〈T 〉 = 〈A〉+ 〈E〉.

2.10.2.8 Evaluating the travel time PDF from the travel time CDF

An alternative technique for the evaluation of the travel time PDF is to solve for the CDF,which can be done by making use of the following forward and backward boundary valueproblems:

∂θG

∂t= −∇ · qG+∇ · θD∇G− qOG in Ω (2.169a)

G(x, 0) = 0 in Ω (2.169b)

[qG(x, t)−D∇G(x, t)] · n = q · n on Γ− (2.169c)

[qG(x, t)−D∇G(x, t)] · n = 0 on Γ0 (2.169d)

for the forward problem, and

∂θG

∂t= ∇ · qG+∇ · θD∇G− qIG in Ω (2.170a)

G(x, 0) = 0 in Ω (2.170b)

[−qG(x, t)−D∇G(x, t)] · n = −q · n on Γ+ (2.170c)

−D∇G(x, t) · n = 0 on Γ0 (2.170d)

for the backward problem. The function G(x, t) is the forward (or backward) travel timeCDF, from which the travel time PDF g(x, t) can be deduced by enforcing:

g(x, t) = ∂G(x, t)∂t

(2.171)

This formulation is more appropriate to account for transient velocity fields. Note thatthe boundary conditions (2.169c) and (2.170c) may be replaced by the first-type conditionG(x, t) = 1. Note also that the n temporal moments µn of the travel time PDF can also bedirectly calculated from the CDF by means of the following formula:

µn(x) =∫ +∞

0n tn−1(1−G(t,x))dt (2.172)


2.10.2.9 Capture zone probability

The capture zone probability of a specific outlet can easily be calculated by adaptingEq. (2.155) for the cdf of the life-expectancy-to-outlet Γn, pn(x, t) =

∫ t0 g(x, u)du:

∂φpn∂t

= ∇ · qpn +∇ · θD∇pn − qIpn in Ω (2.173a)

pn(x, 0) = 0 in Ω (2.173b)

[−qpn(x, t)−D∇pn(x, t)] · n = −q · n on Γn (2.173c)

[−qpn(x, t)−D∇pn(x, t)] · n = 0 on Γ+ (2.173d)

−D∇pn(x, t) · n = 0 on Γ0 (2.173e)

The field of pn(x, t) defines the probabilistic drainage basin corresponding to the outletΓn, or probabilistic capture zone relative to a particular transit time t. The ultimate, orsteady-state probabilistic drainage basin is the field p∞n (x) = pn(x,∞). Because in both2D and 3D domains there is a non-zero probability that the water particles will not beintercepted by the outlet Γn (the only location where this probability is zero), p∞n (x) is lessthan one, p∞n (x) =

∫∞0 g(x, t)dt < 1 ∀x ∈ Ω. An iso-probability value pn(x,∞) includes the

domain for which the fraction 1− pn(x,∞) of its water amounts will reach the outlet Γn,sooner or later.

Chapter 3

Numerical Implementation

3.1 General

HydroGeoSphere uses the control volume finite element method to solve the flow equationsfor all domains considered in a simulation, and it uses either the standard Galerkin finiteelement method or the control volume finite element method to solve the transport equation.Elements available to solve the 3-D porous medium and dual continuum equations arerectangular prisms (8-node elements), 3-D triangular prisms (6-node elements), and 3-Dtetrahedra (4-node elements). The 2-D fracture and surface equations are solved for usingeither rectangular (4-node elements) or triangular elements (3-node elements) and the 1-Dwell, tile drain and channel equations are solved for 1-D linear elements (2-node elements).For the 3-D and 2-D elements, a finite difference approximation is also available, accordingto the method presented by Panday et al. [1993].

The model solves either linear equations (for fully-saturated flow or solute transport) ornon-linear equations (for variably-saturated subsurface flow, surface flow, solute transportwith a flux-limiter, including density-dependent flow and transport). To solve the non-linear equations, HydroGeoSphere uses the robust Newton-Raphson linearization method,except for the weakly nonlinear density-dependent problem, which is solved by the Picardmethod. Although the Newton-Raphson technique requires a larger amount of work foreach solution step compared to other linearization methods such as Picard iteration, therobustness and higher order of convergence of the Newton method make it attractive.

The matrix equation arising from the discretization is solved by a preconditioned iterativesolver, using either the ORTHOMIN, GMRES or BiCGSTAB acceleration.

In this chapter, we present the discretized equations and provide details on the varioussolution schemes used by the model. We first present a general description of the controlvolume finite element method used to discretize the governing equations. We then presentthe discretized equations for subsurface and surface flow and for transport. We finallypresent the solution method for non-linear equations.

55

CHAPTER 3. NUMERICAL IMPLEMENTATION 56

3.2 Control Volume Finite Element Method

The method of solution for the flow problem is based on the control volume finite elementapproach [e.g. Forsyth, 1991] which has been shown to be particularly well-suited for a fastand efficient implementation of the Newton-Raphson linearization technique [Forsyth andSimpson, 1991].

The basic idea of the control volume finite element approach is to obtain a discretized equationthat mimics the governing mass conservation equation locally. A volume of influence, referredto as a control volume, is assigned to each node. The discretized equation for a given nodethen consists of a term describing the change in fluid mass storage for that volume whichis balanced by the term representing the divergence of the fluid mass flux in the volume.The fluid mass flux will depend on the physical properties associated with the volume andthe difference in the value of the primary variable between the node in question and itsneighbors.

Discretization of the subsurface and the surface flow equations is identical except for thedifference in dimensionality. For the sake of clarity, we present here a detailed descriptionof the control volume finite element method applied to discretize a simplified prototypecontinuity equation. The final discretized equations for all subsurface domains and forsurface flow are then presented without providing the details of the derivation.

Let us assume the following prototype flow equation:∂

∂t(θsSw)−∇ · (K · kr∇h) +Q = 0 (3.1)

where h is the hydraulic head, equal to ψ + z.

Let Ni be the standard finite element basis functions such that:

Ni = 1 at node i= 0 at all other nodes∑

j

Nj = 1 everywhere in the solution domain (3.2)

Using the standard basis function, an approximating function is defined in the usual way forthe spatially and temporally variable h and Sw:

h ' h =∑j

Njhj(t)

Sw ' Sw =∑j

NjSwj(t) (3.3)

where j is a nodal index ranging from 1 to n, where n is the total number of nodes.

The standard Galerkin technique (see e.g. Huyakorn and Pinder [1983]) is used to discretizeEquation 3.1 over the domain of interest, V , leading to:∫

V

[∂

∂t(θsSw)−∇ ·

(K · kr∇h

)+Q

]Ni dV = 0 (3.4)


where i is the nodal index ranging from 1 to total number of nodes. We now only consider theequation applying to node i. Upon approximating the time derivative by a finite differencerepresentation and using a lumped mass approach to treat the storage terms in Equation 3.4,we can write: ∫

v

∂

∂t(θsSw)Nidv = θs

(SL+1w − SLw

)∆t

∫vNi dv (3.5)

where v is the region or control volume associated with node i, L is the time level and ∆tis the time-step size. Here, a fully implicit discretization in time is used. Likewise, for theregion v associated with node i, the source/sink term in Equation 3.4 becomes:

Qi =∫vQNi dv (3.6)

It is now desired to express the flux term in Equation 3.4 as a function of the total headdifference between node i and each of its neighbors. By applying the divergence theorem tothis term in Equation 3.4, one obtains:∫

v−∇ ·

(K · kr∇h

)Ni dv =

∫v∇Ni ·K · kr∇h dv −

∫Bq∗Ni dB (3.7)

The last term on the right-hand side is the integral of the fluid flux q∗ normal to the boundary,B, of volume v. Let us assume for clarity that this fluid flux is zero. Use can be made ofEquation 3.3 to get:

∇Ni · ∇h = ∇Ni · ∇

∑j

hjNj

(3.8)

where the summation is carried over all the nodes. Using the relation ∑Nj = 1. we have:

Ni = 1−∑j 6=i

Nj (3.9)

such that:∇Ni = −∇

∑j 6=i

Nj (3.10)

Using Equations 3.9 and 3.10, the right-hand side of Equation 3.8 can now be rewritten inthe following way:

∇Ni · ∇

∑j

hjNj

= ∇Ni · ∇

∑j 6=i

hjNj

+∇ (hjNj)

= ∇Ni · ∇

∑j 6=i

Nj

(hj − hi) (3.11)

The relationship in Equation 3.11 is used to obtain:

∫v∇Ni ·K · kr∇h dv =

∫v∇Ni ·K · kr∇

∑j 6=i

Nj

(hj − hi) dv (3.12)


Discretization of the domain into finite elements will create a nodal connectivity (i.e. atable of nodal incidences for the elements). Defining ηi as being the set of nodes connectedto node i, it is obvious that the nodes not included in ηi will not contribute to the changein storage or fluid flow at node i. Using this result from discretization and the fact thatthe hj − hi are nodal quantities and that the summation and integration operations areinterchangeable, the right-hand side of Equation 3.12 becomes:

∫v∇Ni ·K · kr∇

∑j 6=i

Nj

(hj − hi) dv =∑j∈ηi

∫v∇Ni ·K · kr∇Nj (hj − hi) dv (3.13)

Using the following relation:

γij =∫v∇Ni ·K · ∇Nj dv (3.14)

the right-hand side of Equation 3.13 becomes:∑j∈ηi

∫v∇Ni ·K · kr∇Nj (hj − hi) dv =

∑j∈ηi

λij+1/2γij (hj − hi) (3.15)

where λij+1/2 represent a weighted value of the relative permeabilities for nodes i and j,evaluated at the interface between nodal volumes i and j.

Combining Equations 3.5, 3.6 and 3.15, and using fully implicit time weighting, the finalform of the discretized equation for node i becomes:[

(θsSw)L+1i − (θsSw)Li

] vi∆t =

∑j∈ηi

(λ)L+1(ij+1/2)γij(h

L+1j − hL+1

i ) +QL+1i (3.16)

where superscript L denotes the time level and where the volume of influence for node i isgiven by:

vi =∫vNi dv (3.17)

The discretized equation presented above is independent of the choice of element type. Ofthe numerous types of three-dimensional elements that can be used to discretize the porousblocks, both 8-node rectangular block elements [Huyakorn et al., 1986] and 6-node prismelements are implemented here. The user also has the option of subdividing rectangularblock or prism elements into 4-node tetrahedral elements, which permits the discretizationof highly irregular domains. The two-dimensional fracture planes and the surface flow arediscretized using either rectangular or triangular elements [Huyakorn et al., 1984]. Thischoice of simple elements allows use of the influence coefficient technique [Frind, 1982;Huyakorn et al., 1984] to analytically evaluate the integrals appearing in Equation 3.14 inan efficient manner.

An option that has been implemented in the numerical formulation is a choice between afinite element or a finite difference representation using a methodology identical to thatdescribed by Panday et al. [1993] and Therrien and Sudicky [1996]. The problem and


boundary conditions are defined in terms of finite elements upon input and new nodalconnectivities are established when a finite difference formulation is chosen for both 3-Dblock and prism elements. The influence coefficient matrices for the finite element method,presented by Huyakorn et al. [1986], are manipulated in order to mimic a finite differencediscretization. The reader is referred to Panday et al. [1993] and Therrien and Sudicky[1996] for details on the implementation. The finite difference form of the needed influencecoefficient matrices are provided in Huyakorn et al. [1986]. Because of the different numberof nodal connections (i.e. 7 for finite difference, 27 for finite element for block elements), thefinite element method requires nearly four times as much memory to store the coefficientmatrix compared to that for the finite difference method. For block elements, the size ofthe assembled coefficient matrix is n× 27 for finite elements and n× 7 for finite differences,where n is the number of nodes in the domain. A variety of subsurface flow simulationsthat we have performed for fractured porous media under either fully- or variably-saturatedconditions, have indicated that the finite difference and finite element representations yieldsimilar results. Also, experience has indicated that the CPU time required when using thefinite difference method is also nearly a factor of four less.

In the following section, we present the discretized flow equation for the porous medium,fractures, dual continuum, wells, tile drains and surface. The discretized equations aresimilar to the discretized form of the prototype Equation 3.16 and their derivation, notshown here, follows the steps highlighted in this section.

3.3 Discretized Subsurface Flow Equations

3.3.1 Porous Medium

Using the control volume finite element method, with fully implicit time weighting, thefollowing discretized porous medium flow equation is obtained:[

(SsSwψ + θsSw)L+1i − (SsSwψ + θsSw)Li

] wmvi∆t =

∑j∈ηi


L+1j − hL+1

i )−(∑

ΓL+1ex

)vi ±QL+1

i (3.18)

where:hi = ψi + zi (3.19)

and:γij =

∫v∇Ni · wmK · ∇Nj dv (3.20)

for interpolation functions N defined for the 3-D porous medium elements and where the3-D volume associated with a given node is given by:

vi =∫vNi dv (3.21)


For upstream weighting, the λij+1/2 values are given by:

λij+1/2 = krj if γij (hj − hi) > 0

λij+1/2 = kri if γij (hj − hi) < 0(3.22)

Upstream weighting of the relative permeability is highly recommended in order to ensuremonotonicity of the solution [Forsyth, 1991]. A monotone solution will yield saturationsthat always remain in the physical range, (i.e. between 0.0 and 1.0). Forsyth and Kropinski[1997] provides a vivid illustration of the importance of the type of relative permeabilityweighting on the quality and stability of the solution of Richards’ equation. They considereda two-dimensional unsaturated flow problem in which the air entry pressure was low whichmakes the governing equation more hyperbolic in nature. Results showed that the use ofcentral weighting produced significant negative saturations as the wetting front advanced in arelatively dry soil; however, the solution was stable and remained in the physical range whenupstream weighting was used. Although it is well-known that the use of upstream weightingfor advection-dispersion problems can lead to excessive smearing of a concentration front,its use in conjunction with hyperbolic-type equations, such as the one for variably-saturatedflow, does not smear as much because the solution is self-sharpening. It has also been shownthat, for purely hyperbolic equations, the use of central weighting can cause complete failureof the solution [Sammon, 1988].

The reduction of area available for flow across a fracture-matrix interface can be easilyincorporated in Equation 3.18 in a manner suggested by Wang and Narasimhan [1985]. Thearea between nodes i and j, which are both located in the matrix, is imbedded in the γijterm, defined by Equation 3.20. For cases where i or j also coincide with a fracture node,this area between the two nodes is multiplied by a factor representing the new effective area,accounting for a matrix-matrix flow area reduction when the fracture desaturates.

It should also be noted that in the discretized Equation 3.18, the saturation term isrepresented exactly and no use is made of the water capacity term which is known to inducesevere mass balance errors [Celia et al., 1990, Milly, 1985]. Various schemes ranging incomplexity have therefore been derived to resolve this problematic mass balance error [e.g.Cooley, 1983; Milly, 1985; Celia et al., 1990]. In this work, the use of the Newton-Raphsonprocedure allows a direct representation of the saturation term, thereby completely avoidingthe difficulties arising from the use of the water capacity term.

3.3.2 Discrete Fractures

The discretized 2-D equation for flow in fractures is:[(Swf )L+1

i − (Swf )Li] wfai

∆t =∑j∈ηf i

(λf )L+1(ij+1/2)γf ij(hf

L+1j − hfL+1

i ) + wfΓL+1f ai (3.23)

where ηf i is the set of fracture nodes connected to fracture node i through the 2-D fractureelements and where:

hf i = ψf i + zf i (3.24)


and:γf ij =

∫a∇Ni · wfKf · ∇Nj da (3.25)

with interpolation functions N defined for the 2-D fracture elements.

The 2-D area associated with a given fracture node is given by:

ai =∫aNi da (3.26)

For upstream weighting of relative permeabilities, the (λf )ij+1/2 values are given by:

(λf )ij+1/2 = krfj if γf ij

(hf j − hf i

)> 0

(λf )ij+1/2 = krfi if γf ij

(hf j − hf i

)< 0

(3.27)

3.3.3 Dual Continuum

Using the control volume finite element method, the discretized equation for flow in a dualcontinuum is: [

(SsdSwdψd + θsdSwd)L+1i − (SsdSwdψd + θsdSwd)Li

] wdvi∆t =

∑j∈ηid

(λd)L+1(ij+1/2)γdij(hdj

L+1 − hdiL+1)−[ΓL+1d +Qdi

L+1]vi (3.28)

where volume vi is defined for the dual continuum nodes, with an expression similar toEquation 3.21 and where:

hdi = ψdi + zdi (3.29)

and:γdij =

∫v∇Ni · wdKd · ∇Nj dv (3.30)

where the interpolation functions N are defined for the 3-D dual continuum elements.

For upstream weighting of relative permeabilities, the λdij+1/2 values are given by:

λdij+1/2 = kdrj if γdij(hdj − hdi

)> 0

λdij+1/2 = kdri if γdij(hdj − hdi

)< 0

(3.31)

3.4 Discretized Surface Flow Equation

The discretized 2-D surface flow equation is:[(ho)L+1

i − (ho)Li] ai

∆t =∑j∈ηoi

(λo)L+1(ij+1/2)γoij(ho

L+1j − hoL+1

i ) + doΓL+1o ai ± qoi (3.32)


where ηoi is the set of surface nodes connected to surface node i through the 2-D surfaceelements and where:

hoi = doi + zoi (3.33)and:

γoij =∫a∇Ni ·Ko · ∇Nj da (3.34)

with interpolation functions N defined for the 2-D surface elements.

The 2-D area associated with a given surface node is given by:

ai =∫aφoNi da (3.35)

For upstream weighting of surface pseudo relative permeabilities, the (λo)ij+1/2 values aregiven by:

(λo)ij+1/2 = kroj if γoij(hoj − hoi

)> 0

(λo)ij+1/2 = kroi if γoij(hoj − hoi

)< 0

(3.36)

Careful consideration should be provided to implementing the conductances of the flowterms of Equation 3.34. The conductance tensor Ko has two components in 2 dimensions,Kox and Kox, which are combinations of the properties of the two nodes involved in therespective flow connection. The constant part in Equations 2.42 and 2.43, constitutes thefrictional resistance term (1/nx and 1/ny for Manning, Cx and Cy for Chezy, or

√8g/fx and√

8g/fy for Darcy-Weisbach, all referred to as Hx and Hy) and is an input parameter foreach element. The gradient part of Equations 2.42 and 2.43, [∂ho/∂s]−1/2, is calculated fromthe average x- and y-direction gradients of the connecting cells, to determine the gradient inthe direction of maximum slope. The remaining terms in Equations 2.42 and 2.43, along withthe depth within the first gradient operator of Equation 2.45, combine to d5/3

o for Manning,d

3/2o for Chezy and d

3/2o for Darcy-Weisbach. Full upstream weighting of this term between

the two connecting nodes ensures a monotonic solution, without unphysical oscillations.Further, upstream weighting ensures that flow from a dry node is zero, maintaining thephysical reality to the set of governing equations.


One-dimensional hydraulic features in HydroGeoSphere are used to simulate streams,rivers, wells, water supply lines and gravity driven drains. Similar to surface/subsurfaceflow, the 1D equations were discretized with a control volume finite element approachcombined with the Newton-Raphson successive linearization method to solve the discreteset of nonlinear equations.

The governing equation for general one-dimensional flow can be written as:

L(h1D) = AfCs∂h1D∂t

+ ∂Af∂t− ∂

∂s

(K1D

∂h1D∂s

)−Qwδ(s− sp)− Γexch (3.37)


K1D(h1D) = C ·Af · (RH)p ·[∂h1D∂s

]q−1(3.38)

where the change in water volume is expressed as a function of the compressibility coeffi-cient (Cs) and the change in the flow area. The fluid exchange between the surroundingenvironment and hydraulic feature is represented by Γexch.

The finite element analysis utilizes a trial solution for head, defined using the standardinterpolation functions and then a weighted residual at node i can be defined as the weightedintegral of the equation over the solution domain such that:

h1D =∑j

Nj(h1D)j (3.39)

∫R1NiL(h1D)dR1

=∫R1Ni

[AfCs

∂h1D∂t

+ ∂Af∂t− ∂

∂s

(K1D

∂h1D∂s

)−Qwδ(s− sp)− Γexch

]dR1

= 0

(3.40)

Introducing the trial solution and using a lumped capacitance formulation:

Li

[Af,iCs

∂h1D,i∂t

+ ∂Af,i∂t

]

=∫R1Ni

∂∂s

K1D∂

∂s

∑j

Njh1D,i

+Qwδ(s− sp) + Γexch

dR1

= −∑j

h1D,j

[∫R1K1D

∂Ni

∂s

∂Nj

∂sdR1

]− qδR1,i +Qw,i + Γexch,i

(3.41)

where:

Li

[Af,iCs

∂h1D,i∂t

+ ∂Af,i∂t

]≈∫R1Ni

[AfCs

∂h1D∂t

+ ∂Af∂t

]dR1 (3.42)

qδR1,i = −∫δR1

Ni

(K1D

∂h1D∂s

)d(δR1) (3.43)

Qw,i =∫R1NiQwδ(s− sp)dR1+

∫R1NiΓexchdR1 (3.44)

Γexch,i =∫R1NiΓexchdR1 (3.45)

Incorporating the properties of standard interpolation functions ∑iNj = 1 or Ni = 1 −∑j 6=iNj , the discrete control volume one-dimensional flow equation is converted to the

following form:

Li

[Af,iCs

∂h1D,i∂t

+ ∂Af,i∂t

]=∑j 6=i

C1D,ij [h1D,j − h1D,i]− qδR1,i +Qw,i + Γexch,i (3.46)


where:

C1D,ij =∫R1K1D(hup,ij)

∂Ni

∂s

∂Nj

∂sdR1 (3.47)

where the hydraulic properties (K1D) are determined by the upstream weighting methoddetermined by hup,ij = max (h1D,i, h1D,j).

3.6 Flow Coupling

The porous medium (and the dual continuum if present) is discretized in three dimensionswith either rectangular prisms, triangular prisms or tetrahedra. Two-dimensional rectangularor triangular elements represent the discrete fracture and the surface domains, and one-dimensional line elements represent the wells and the tile drains. When discretizing thedomain, nodes forming the fracture or surface elements or nodes forming the well and tiledrain elements have to coincide with those on the adjacent porous medium finite volumes,similarly to Sudicky et al. [1995]. Because the fracture elements are generated such that theycorrespond to planes that intersect three or 4 nodes in the three-dimensional rectangularblocks or prisms, the nodes comprising the fracture elements are therefore common to nodescomprising the porous matrix elements. The commonality of these nodes thus ensures thecontinuity of hydraulic head at the fracture matrix interface. Also, by superimposing thecontributions at each node from both element types, there is no need to explicitly calculatethe fluid leakage terms appearing in the discretized equations. For the surface flow nodes, achoice of common or dual nodes is offered, but the surface flow nodes nevertheless have tocoincide with porous medium nodes. Additionally, nodes forming the 3-D dual continuumdomain coincide with the porous medium nodes, but they form duplicate nodes since thedual node approach is used.

With the common node approach, the matrix contributions arising from the discretization ofthe discrete fracture or the surface nodes, as well as matrix contributions from the well nodesand tile drain nodes are superimposed onto those stemming from the discrete form of porousmedium equation. Continuity in pressure head is therefore ensured between the differentdomains, which avoids the need for a direct evaluation of the exchange fluxes between theporous medium elements and the other domains.

As an example, we write below the final discretized matrix equation when discrete fracturesare located in the porous domain. Using discretized Equation (3.23), we write the couplingterm Γf as:

wfΓL+1f ai =

[(Swf )L+1


∆t − wf∑j∈ηf i


L+1j − hfL+1

i ) (3.48)

Replacing the expression for the coupling term into porous medium Equation 3.18 gives the


following global equation:[(SsSwψ + θsSw)L+1

i − (SsSwψ + θsSw)Li] wmvi

∆t +[(Swf )L+1


∆t =

∑j∈ηi


L+1j − hL+1

i )−(∑

ΓL+1ex

)vi ±QL+1

i +

wf∑j∈ηf i


L+1j − hfL+1

i ) (3.49)

In Equation 3.49, we assume that hi = hf i for node that are common to the porous mediumand fracture domains, which ensures continuity. The exact value of the fluid exchangebetween the domains, Γf , is therefore not computed explicitly prior to solution, but theexchange can be back-calculated during post-processing of results by evaluating Equation 3.48at the desired nodes.

Similarly to the approach shown here for superposition of 2 domains, the model allowssuperposition of all flow domains by adding the relevant discretized equations and assumingcontinuity of head.

In the next section, we present coupling for the dual node approach when subsurface andsurface flow equations are fully coupled. It is the only approach currently available forcoupling the porous medium and second subsurface continuum and it is one of the twooptions available for coupling the porous medium and surface flow.

3.6.1 Dual-Continuum Subsurface Coupling

As mentioned previously, the porous medium equation is discretized everywhere for the 3-Dsimulation grid. On the other hand, the dual continuum need not be present everywhere in thedomain. When dual continuum is simulated, the discretized dual continuum Equation 3.28is applied only to those grid nodes where dual continuum conditions exist. Because thedual node approach is used to link the porous medium and the second continuum, thedual continuum nodes have the same spatial coordinates than the porous medium nodes.However, at a dual node, there are two unknowns to solve for: the total hydraulic head inthe porous medium (ψ + z) and that in the second continuum (ψd + zd). Fluid exchangebetween the two domains is provided by the exchange term Ωd.

After the subsurface flow equations have been assembled into an implicit system of matrixequations, the 3-D dual continuum flow equations are added to the matrix along with theirporous medium interactions, and the fully-integrated flow system is solved at each timestep.

3.6.2 Surface - Subsurface Coupling

The 2-D areal surface flow modules of HydroGeoSphere follow the same conventionsfor spatial and temporal discretizations as those used by the subsurface modules. The


Figure 3.1: Spatial Discretization of the Surface Flow System and its Connection to theSubsurface.

surface flow equation is solved on a 2-D finite-element mesh stacked upon a subsurfacegrid when solving for both domains (i.e. the x- and y-locations of nodes are the same foreach layer of nodes), as shown in Figure 3.1. For superposition, the grid generated for thesubsurface domain is mirrored areally for the surface flow nodes, with surface flow nodeelevations corresponding to the top elevation of the topmost active layer of the subsurface grid.Note that surface flow node elevations may vary substantially to conform with topography.However, the assumptions of small slope inherent in the diffusion-wave equation will notallow for modeling of inertial effects.

Although Figure 3.1 uses 2-D rectangular elements on the surface and 3-D hexahedralelements in the subsurface, HydroGeoSphere allows for the use of unstructured gridscomposed of, for example, 2-D triangles on the surface and 3-D 6 node prismatic or 4-nodetetrahedral in elements in the subsurface.

The discretized surface Equation 3.32 is coupled with the 3-D subsurface flow Equation 3.18via superposition (common node approach) or via leakage through a surficial skin layer(dual node approach). For both approaches, fully implicit coupling of the surface andsubsurface flow regimes provides an integral view of the movement of water, as opposedto the traditional division of surface and subsurface regimes. Flux across the land surfaceis, therefore, a natural internal process allowing water to move between the surface andsubsurface flow systems as governed by local flow hydrodynamics, instead of using physicallyartificial boundary conditions at the interface.


When the subsurface connection is provided via superposition, HydroGeoSphere adds thesurface flow equation terms for the 2-D surface mesh to those of the top layer of subsurfacenodes in a manner similar to that described in Section 3.6 for the common node approach.In that case, the fluid exchange flux Γo, which contains leakance term Kso does not need tobe explicitly defined.

For subsurface connection via skin leakance, the dual node approach is used and each surfaceflow node communicates with the first active subsurface flow node directly beneath it toform the subsurface connection. The subsurface flow connection term Kso must therefore beadded to the left hand side of the respective row for corresponding subsurface flow nodes,in the column that connects them to the surface flow nodes. The interaction flux will alsobe added to the right hand side of the subsurface flow Equation 3.18. After the subsurfaceflow equations have been assembled into an implicit system of matrix equations, the 2-Dareal surface flow equations are added to the matrix along with their subsurface interactions,and the fully-integrated flow system is solved at each time step. This provides significantrobustness and accuracy over flux linkage techniques, often used when conjunctive modelingis required for the surface and subsurface regimes.

3.7 Flow Boundary Conditions

3.7.1 Subsurface Flow

Boundary conditions for subsurface flow include the following: first-type (Dirichlet) bound-aries of prescribed hydraulic head, areal infiltration or recharge, source/sinks, evapotranspi-ration and seepage faces.

The source/sink term in Equation 3.18 can be manipulated in order to impose prescribed headboundary conditions [Forsyth, 1988; Forsyth and Kropinski, 1997]. To assign a prescribedpressure head, ψb, to a portion of the domain, the source/sink term becomes:

Qi = KijkrwWi(ψb − ψi) (3.50)

where Wi is a number large enough (e.g. 1020) to ensure that ψi = ψb when the assembledsystem of equations for all nodes is solved. This can be viewed as injecting or withdrawingsufficient fluid at node i to maintain the prescribed pressure head.

Seepage faces represent boundaries that require special treatment and a variety of methodshave been suggested to implement such boundaries [Neuman, 1973; Cooley, 1983]. Themethod used here is from Forsyth [1988] and requires that the approximate location of theseepage face be known a priori. The appropriate form of the source/sink term at the nodesforming the seepage face will be:

Qi = K∗krwWi (ψatm − ψi) ψi > ψatm

= 0 ψi < ψatm (3.51)

where K∗ is the component of the hydraulic conductivity tensor normal to the seepage face.


The above expression allows seepage only when the pressure in the medium is greater thanthe atmospheric pressure, ψatm.

A free drainage boundary is often used for unsaturated flow conditions. It assumes thata unit hydraulic gradient exists along the vertical direction, which results in the followingvolumetric flow rate out of the domain:

Qi = KzzkrwAi (3.52)

where Ai is the outflow area associated with node i where free drainage occurs.

Drain nodes can be used to simulate fluid flow out of the domain, but without allowinginflow. For a given drain node i, the drain flow rate is given by

Qi = CDR(hi − hDR) hi > hDR

Qi = 0 hi ≤ hDR (3.53)(3.54)

where CDR is an equivalent conductance [L2 T−1] for the drain node and hDR is the drainhydraulic head.

River nodes can be used to simulate fluid flow into or out of the domain. For a given rivernode i, the flow rate is given by:

Qi = CRIV (hi − hRIV ) (3.55)

where CRIV is an equivalent conductance [L2 T−1] for the river node and hRIV is theriver hydraulic head. Assuming the river bed acts as a semipervious layer, the equivalentconductance CRIV is given by:

CRIV = KBED

LBEDWRCHLRCH (3.56)

where KBED is the river bed conductivity [L T−1], LBED is the river bed thickness [L],WRCH is the width of the reach [L] and LRCH is the length of the reach [L].

3.7.2 Surface Flow

Boundary conditions to the surface flow system include the following: first-type (Dirichlet)boundaries of prescribed water elevation, rainfall rate, source/sinks, evaporation, zero-depthgradient and critical-depth conditions.

First-type boundary conditions are implemented in an identical manner to what is describedfor subsurface flow in Section 3.7.1 above. Areal rainfall (volumetric inflow over an area) isimplemented as an input water flux multiplied by the contributing area. Sources/Sinks areapplied as net fluxes to the surface flow nodes that receive them. Sinks are constrained bythe physical property that water depth cannot be negative (i.e. water cannot be extractedwhen the water level is below bed elevation). If this condition occurs at any solution iteration,


only as much water is withdrawn as to not violate this constraint. A further constraintis that injection should also be restricted at sink nodes. Thus, the sink strength shouldonly reduce to zero under limited supply conditions and not become a negative sink (i.e. asource). Evaporation is applied as an areal sink to an surface flow node, subject to similarnon-negative depth constraints as discussed for sinks, above.

Zero-depth gradient and critical depth boundary conditions are implemented to simulateconditions at the lower boundaries of a hill slope. Zero-depth gradient (ZDG) conditionforces the slope of the water level to equal the bed slope which is provided by the user atthis boundary. The discharge, Qo, at the zero-depth gradient boundary is given for theManning equation by:

Qo = 1nid5/3o

√So (3.57)

for the Chezy Equation by:Qo = Cid

3/2o

√So (3.57)

and for the Darcy-Weisbach relation by:

Qo =√

8gfid3/2o

√So (3.57)

where Qo is the flux per unit width, i is the direction of the zero-depth gradient discharge(i = x in the x-direction and i = y in the y-direction). ni is Manning roughness in thedirection i, Ci is the Chezy coefficient in direction i, fi is the friction factor along directioni, and So is the bed slope at the zero-depth gradient boundary.

Critical depth (CD) condition forces the depth at the boundary to be equal to the criticaldepth. The discharge Qo per unit width at the critical depth boundary is given by:

Qo =√gd3

o (3.58)

3.7.3 Interception and Evapotranspiration

3.7.3.1 Interception

For each time step ∆t, the actual interception storage Sint is calculated as follows:

Ecan = min(Ep, SMax

int /∆tcan, Pp + S0int/∆t), ∆tcan > 0

min(Ep, Pp + S0int/∆t), otherwise

(3.59)

S∗int = S0int + (Pp − Ecan)∆t (3.60)

Sint = min(SMaxint , max(0, S∗int)) (3.61)

where S0int is the value of Sint [L] at the previous time, S∗int is the intermediate value of Sint

[L], Pp is the precipitation rate [L T−1], Ecan is the canopy evaporation [L T−1], Ep is thereference evapotranspiration [L T−1], and ∆tcan is the canopy evaporation interval [T].


Ep may be derived from pan measurements or computed from vegetation and climaticfactors (radiation, wind, humidity, and temperature) using the Penman-Monteith equation[Monteith, 1981] for vegetated surfaces or a bare-ground evaporation formula [Sharika et. al.,2000] for non-vegetated surfaces, as a function of temperature, wind, and humidity conditions.The reference evapotranspiration is computed efficiently at the start of a simulation, forfurther use in determining the actual evapotranspiration. The reduced rainfall rate whichreaches ground surface after interception is given by:

PEp = max[0, Pp −(Sint − S0

int)∆t − Ecan] (3.62)

where PEp is the effective rainfall rate [L T−1].

Similarly, the potential for evapotranspiration from the soil surface and below is reduced bythe canopy evapotranspiration term, which is carried through in the subsequent discussions.

3.7.3.2 Evapotranspiration

Evapotranspiration affects elements and nodes in both surface and subsurface flow domains,and may involve several nodal layers. The rate of transpiration for node i (Tpi) can beestimated by substituting the nodal water content θi and nodal time-varying root distributionfunction RDFi in Equations 2.74 to 2.79.

Equation 2.76 is obtained by expressing Equation 2.78 over the discretized layers of elementscontaining the roots. As a practical matter, however, the value of RDF at any areal locationmay be less than one to account for ineffective roots. For each time step, the transpiration(Tp) over the effective root length is calculated as the sum of the transpiration from each ofthe nodes at depth, as

Tp =nR∑i=1

Tpi (3.63)

where nr is the number nodes that lie within the depth interval 0 ≤ z ≤ Lr, at any areallocation.

The rate of evaporation for node i (Esi) can be estimated by substituting the nodal watercontent θi and nodal evaporation distribution function EDFi in Equations 2.81 to 2.85.

An appropriate EDFi for each cell layer may be prescribed as a function of its depth fromland surface.

3.8 Elemental Velocities

Elemental velocities or fluid fluxes can be computed after solving for fluid flow. The fluidflux is evaluated by combining its definition:

q = −K · kr∇(ψ + z) (3.64)


with the definition of the interpolation function for the unknown. For pressure head, theinterpolation function is:

ψ =∑j

Njψj(t)

The evaluation of elemental velocities or fluid fluxes is done according to the simple formulasgiven by Huyakorn et al. [1986] for all types of 1-D, 2-D and 3-D elements available in themodel.

3.9 Discretized Solute Transport Equations

3.9.1 Porous Medium

When the transport equation is linear, which is the case except when a flux limiter is usedfor the advective term in order to minimize adverse numerical dispersion, its solution isnot as involved as that for the non-linear cases. We present here a discretization schemefor the transport equations based on the control volume finite element method presentedin Section 3.2. The type of elements used for transport are identical to those used for theflow problem and the choice of superposition of several domains (common node approach)is also given, where elements representing one domain are superimposed onto the elementrepresenting a second domain, as performed for the flow equation. This ensures the continuityof concentration at the domain-to-domain interface and avoids the need to explicitly determinethe solute mass exchange terms involving Ωex in the governing transport equations. When adual continuum is simulated, the dual node approach is used to represent the interactionbetween the porous medium and the dual continuum.

We present here the standard discretized equation for the 3-D porous medium, obtainedfrom the application of the control volume finite element method:[

(θsSwRC)L+1i − (θsSwRC)Li

] wmvi∆t =

∑j∈ηi

(C)L+1(ij+1/2)(λ)L+1

(ij+1/2)γij(hL+1j − hL+1

i )+

∑j∈ηi

χij(CL+1j − CL+1

i ) + (QiCups)L+1i +

[(RλCi)par − (θsSwRλC)i +

∑ΩL+1

ex

]vi (3.65)

where:Cups = Ci if Qi < 0

= Cinflow if Qi > 0 (3.66)

and where Cinflow is the specified source inflow concentration (recall that Qi is the source/sinkterm used to represent boundary conditions). Note that qL+1

(ij+1/2) = (λ)L+1(ij+1/2)γij(h

L+1j −

hL+1i ) is the fluid flux at the interface between nodes i and j and is back-calculated from

the flow solution.

The term C(ij+1/2) in Equation 3.65 depends on the type of advective weighting used. Forcentral weighting:

C(ij+1/2) = Ci + Cj2 (3.67)


Upstream weighting gives:

C(ij+1/2) = Cups = Cj if γij (hj − hi) > 0

C(ij+1/2) = Cups = Ci if γij (hj − hi) < 0(3.68)

A TVD type flux limiter is also available to evaluate C(ij+1/2) according to [Van Leer, 1974;Unger et al., 1996]:

C(ij+1/2) = Cups + σ(rij)(Cdwn − Cups

2

)(3.69)

where Cdwn is the concentration of the downstream node between i and j. The smoothnesssensor rij is given by:

rij =(Cups − Ci2ups‖Pups − Pi2ups‖

)(Cdwn − Cups‖Pdwn − Pups‖

)−1

(3.70)

where Ci2ups is the second upstream node between i and j, and Pups, Pi2ups, Pdwn are theposition vectors of the upstream, second upstream and downstream nodes, respectively.

A van Leer flux limiter is used in Equation 3.69 such that:

σ(r) = 0 if r ≤ 0

σ(r) = (2r)/(1 + r) if r > 0(3.71)

We further define:χij = −

∫v∇Ni · θsSN+1

w D · ∇Nj dv (3.72)

Other terms in Equation 3.65 are as defined for the discrete flow Equation 3.18.

3.9.2 Discrete Fractures

The discretized 2-D transport equation in fractures is:[(SwfRfCf )L+1

i − (SwfRfCf )Li] wfai

∆t = wf∑j∈ηf i

CfL+1(ij+1/2)λf

L+1(ij+1/2)γf ij(hf

L+1j − hfL+1

i )+

∑j∈ηf i

χfij(CfL+1j − CfL+1

i ) +[(RfλCf i

)par− (SwfRf λCf )i + ΩL+1

f

]ai (3.73)

where:χfij = −

∫a∇Ni · SN+1

wf Df · ∇Nj da (3.74)

and where the interface permeability can be either central or upstream weighted or given bya TVD flux limiter, as shown for the porous medium equation in Section 3.9.1. Other termsin Equation 3.73 are as defined for the discrete fracture flow Equation 3.23.


3.9.3 Double Porosity

The discrete equation for transport into the immobile region of a double-porosity domain isgiven by: [

(θImmCImm)L+1i − (θImmCImm)Li

] vi∆t = ΩL+1

Immvi (3.75)

3.9.4 Isotopic Fractionation

The discrete equation for isotopic fractionation into the immobile (solid) region is given by:

[(CImm)L+1

i − (CImm)Li] vi

∆t = ΩL+1Immxr

vi (3.76)

3.9.5 Dual Continuum

For the dual continuum, the discretized 3-D transport obtained after using the controlvolume finite element method is:[(θsdSwdRdCd)L+1

i − (θsdSwdRdCd)Li] wdvi

∆t =∑j∈ηdi

(Cd)L+1(ij+1/2)(λd)

L+1(ij+1/2)γdij(hd

L+1j −hdL+1

i )+

∑j∈ηdi

χdij(CdL+1j − CdL+1

i ) + (QdCups)L+1i +

[(RdλdCdi)par − (θsdSwdRd λdCd)i + ΩL+1

d

]vi

(3.77)

All terms in Equation 3.77 are defined in a manner analogous to the discretized porousmedium transport equation shown in Section 3.9.1.

3.9.6 Wells

One-dimensional transport along a well is described by the following discretized equation:[(Cw)L+1

i − (Cw)Li] πr2

s li∆t =

∑j∈ηwi

(Cw)L+1(ij+1/2)(λw)L+1

(ij+1/2)γwij(hwL+1j − hwL+1

i )+

∑j∈ηwi

χwij(CwL+1j − CwL+1

i ) + (QwCups)L+1i + πr2

s

[(λCwi)par + ΩL+1

w

]li (3.78)



3.9.7 Tile Drains

One-dimensional transport in a tile drain is described by the following discretized equation:[(Ct)L+1

i − (Ct)Li] Ali

∆t =∑j∈ηti

(Ct)L+1(ij+1/2)(λt)

L+1(ij+1/2)γtij(ht

L+1j − htL+1

i )+

∑j∈ηti

χtij(CtL+1j − CtL+1

i ) + (QtCups)L+1i +

[A (λCti)par +AΩL+1

t

]li (3.79)


3.9.8 Surface runoff

For the surface flow domain, the governing transport equation is discretized as:[(φohoRoCo)L+1

i − (φohoRoCo)Li] ai

∆t =∑j∈ηoi

CoL+1(ij+1/2) · qo

L+1(ij+1/2)

+∑j∈ηoi

χoij(CoL+1j − CoL+1

i ) +[(φohoRoλCoi)par − (φohoRoλCo)i + ΩL+1

o

]ai (3.80)

where:χoij = −

∫a∇Ni · (φoho)N+1Do · ∇Njda (3.81)

and where the interface flux is obtained from the solution to the associated flow equation.The term Co

L+1(ij+1/2) may be treated in a mid-point or upstream weighted manner, or by

using the TVD flux limiter as discussed earlier in Section 3.9.1.

3.9.9 Channels

One-dimensional transport in a channel is described by the following discretized equation:[(Cc)L+1

i − (Cc)Li] Ali

∆t =∑j∈ηci

(Cc)L+1(ij+1/2)(λc)

L+1(ij+1/2)γcij(hc

L+1j − hcL+1

i )+

∑j∈ηci

χcij(CcL+1j − CcL+1

i ) + (QcCups)L+1i +

[A (λCci)par +AΩL+1

c

]li (3.82)



3.9.10 Thermal Energy

The numerical implementation of variably-saturated subsurface thermal energy transport issimilar to that given by Graf (2005). The discrete form of equation 1 is given by:

Nn∑J=1

TL+1J

∑e

∫V e

(ρbcb∆t wIwJ + (qiρwcw)wI

∂wJ∂xi

+ (kb + ρbcbDij)∂wI∂xi

∂wJ∂xj

)dV e

=Nn∑J=1

TLJ

∑e

∫V e

ρbcb∆t wIwJdV

e

+

Nn∑J=1

∑e

∮Γe

((kb + ρbcbDij)wI

∂T

∂n

)dΓe

+∑e

∫V e

(±QT )wIdV e +∑e

∫Γb

ΩowIdΓb

i, j = 1, 2, 3 (3.83)

where Nn represents the total number of subsurface nodes, L represents the time level atwhich a solution is known and L+ 1 represents the time level for which a solution is desired,∑e represents the sum over all porous media elements connected to node J , Ve represents

the elemental volume, wI and wJ represent the linear basis functions following the Galerkinapproach, Γe represents the elemental boundary with n as the unit vector normal to theboundary, and Γb represents the elemental boundary between the surface and subsurfacedomains.

The numerical implementation of the surface thermal energy transport equation is similarto that for the subsurface; however, the surface equation is depth-averaged and is thusevaluated over the elemental area (Ae), not volume. In addition, the surface thermal energytransport equation includes Eatm representing the atmospheric inputs to the surface thermalenergy system. The discrete equation for the surface regime is given by:

Non∑J=1

TL+1oJ

∑eo

∫Ae

(ρwcwdo

∆t wIwJ + (qoiρwcw)wI∂wJ∂xi

+ (kw + ρwcwDoij)do∂wI∂xi

∂wJ∂xj

)dAe

=Non∑J=1

TLoJ

∑eo

∫Ae

(ρwcwdo

∆t wIwJ

)dAe

+

Non∑J=1

∑eo

∮Γe

((kw + ρwcwDoij)dowI

∂T

∂n

)dΓe

+∑eo

∫Ae

(Eatm ±QT )wIdAe +∑eo

∫Γb

ΩodowIdΓb

i, j = 1, 2 (3.84)

where Non represents the number of overland flow nodes, and eo represents overland elements.


Since the time-Dirac delta function δ(t) cannot properly be handled in the time domain, thePDF’s are evaluated by solving initial value problems (cf Eqs. (2.159) and (2.160)). Thefollowing summarizes the main operations implemented in FRAC3DVS:


1. The age/life expectancy/travel time are taken as specific non-reactive species.

2. The input mass is controlled by nodal influence volumes (times mobile water content),according to Eq. (2.159b) or (2.160b). The injection point xi corresponds to any pointin the system for a travel time evaluation from xi to x, to any point on the rechargearea Γ− for the age problem, or to any point on the discharge area Γ+ for the lifeexpectancy problem.

3. The travel time probability can be calculated:

• For each element: q(x, t) and g(x, t) are evaluated at the element centroid, usingelement shape functions. The control plane section area A(x) is evaluated byintersecting the plane normal to velocity with the corresponding element (seeFig. 3.2).

• At each node: The Super-Convergent Patch Recovery method (SCPR, see[Zienkiewicz, 1992, Wiberg, 1997]) is used to evaluate the nodal velocity (seeFig. 3.3). The fundamental basis of the SCPR method is that there are specificlocations in an element where the derivatives are more accurate for a givenpolynomial degree. These particular locations are referred to as element Super-Convergent points. The SCPR method can briefly be summarized by the following:

– For a given node, a patch is made of neighboring elements (1D, 2D, 3D);– Velocity is evaluated at some sample points within the patch. These sample

points correspond to a set of Gauss points;– A least-square fit is performed through these samples, and the local problem

is solved with a Singular Value Decomposition Factorization solution method.

4. Travel time statistics are post-processed for each element (or node).

3.11 Solute Transport Coupling

Coupling of the various domains for solute transport is done in a manner similar to thecoupling used for fluid flow. When the common node approach is used, the discretizedequations for the coupled domains are added, and continuity of concentration is assumedat the nodes shared by the coupled domains. The exchange term Ωex does not need to beexplicitly evaluated but can be back-calculated after solution.

For the dual node approach, currently available for coupling between the porous mediumand dual continuum domains, the exchange term is explicitly evaluated and there is noassumption of equilibrium or continuity between the concentrations of the two domains. Forthe dual continuum approach, the coupling term is evaluated according to:

ΩL+1d vi =

[−(umC)L+1

i − (udCd)L+1i

]vi (3.85)


Figure 3.2: Procedure for the evaluation of the travel time PDF at the element centroid, forthe case of a brick element.

Figure 3.3: Schematic illustration of SCPR method. Left: Four quadrangles defining a patchof elements used to evaluate the velocity at a node (red circle), with indicated sample points(+); Right: Example of a patch made of 8 brick elements.


When the dual-node approach is used to couple the surface and subsurface domains, thecoupling term may be expressed as:

ΩL+1o = CL+1

ups Γo (3.86)

where CL+1ups = CL+1

o when the flux is from the surface to the subsurface system andCL+1ups = CL+1 when the flux is from the subsurface to the surface system.

3.12 Solute Transport Boundary Conditions

3.12.1 Subsurface Transport

Boundary conditions for transport include the following: first-type (Dirichlet) boundaries,second type (mass flux), third type (total flux) and each can be input as time-dependentquantities.

Some input fluxes and boundary conditions for thermal energy transport were previouslyincorporated into HydroGeoSphere by Graf (2005), such as the prescribed temperatureboundaries. Other thermal energy inputs were developed during this research; specificallythe temperature flux input and atmospheric inputs.

The temperature input flux is incorporated into HydroGeoSphere to account for thethermal energy associated with incoming fluid flux. The temperature input flux can be usedin both the surface and subsurface domains. The general equation is given as:

Qheati = Q(ρincinTin − ρiciTi) (3.87)

where Qheati is the temperature flux into node i, Q is the volumetric flux of the carrier fluid,ρ is the density, c is the heat capacity, T is the temperature, in denotes the input term, andi denotes a nodal term. The density and heat capacity terms vary depending on the materialassociated with the node or input flux (bulk or water). A temperature flux associated withsurface water entering the surface domain (for example, a stream inlet) would use ρ and cterms for water for both the input condition (incoming surface water), and the surface nodereceiving the incoming fluid. However, the injection of heated water into the subsurfacewould use ρ and c terms for water for the injected fluid, but would use bulk values for thesubsurface node.

The atmospheric temperature inputs implemented into HydroGeoSphere include pre-scribed values (i.e. incoming shortwave radiation, air temperature, cloud cover, etc.) orvalues determined by a sinusoidal input function. The addition of atmospheric inputswhich can vary sinusoidally with time allows for the representation of diurnal fluxes for airtemperature, shortwave radiation and wind speed.


3.13 Numerical Techniques

3.13.1 Matrix Solution

Discretized flow equations, such as Equations (3.18), (3.23), (3.28), (3.32), (3.46), or anycombination of these equations, forms a matrix system of the form:

Ax = b (3.88)

where x is the unknown, A is a matrix of coefficients and b is a force vector. When flow isfully-saturated, the discretized equations are linear and the matrix of coefficient A is alsolinear. A direct solution of the matrix equation is then possible. The resulting system ofequations can be very large for a fully three-dimensional field-scale problem. A fast andefficient matrix solver is therefore critical in order to reduce core memory requirements andCPU time. The preconditioned Krylov subspace iterative solver has been shown to be veryefficient and robust for solving large systems of equations [Behie and Forsyth, 1984] andit has therefore been implemented to solve the system of equations. The preconditioningchosen consists of performing an ILU decomposition of the assembled coefficient matrixwithout altering its original sparsity pattern [Behie and Forsyth, 1984]. The solver has thecapability, however, to perform a higher-order ILU decomposition of the coefficient matrix,where additional steps of a Gaussian elimination are performed, resulting in the additionof extra bands to the original matrix. Performing a higher-order decomposition results ina better-conditioned decomposed matrix, which can improve the convergence rate of thesolver, but at the expense of increased storage requirements and additional computationtime.

An option to use either a finite difference or finite element discretization, as describedearlier, has also been implemented for the transport solution. Experience has indicated thatfor discretely-fractured porous media in which the matrix has low permeability such thatmechanical dispersion in the matrix is weak relative to molecular diffusion, the finite elementand finite difference representations give essentially identical results. This suggests that thecross-derivative terms in the transport equation are small compared to the terms that areretained in the finite difference approach.

As in the case of the flow problem, a mass balance is performed to assess the accuracy ofthe solution. A procedure similar to that described by Huyakorn and Pinder [1983] is used.The transport matrix equations are solved using the same ILU-preconditioned ORTHOMINsolver [Behie and Forsyth, 1984] as is used for the flow problem.

3.13.2 Newton-Raphson Method

The Newton-Raphson technique is used to linearize the non-linear equations arising fromdiscretization of variably-saturated subsurface or surface flow equations. The methodis described in Huyakorn and Pinder [1983] and is reproduced here to demonstrate theadvantage of using the control volume finite element approach over a conventional Galerkin


method. To illustrate the method, we apply it to Equation 3.16, which is rewritten in thefollowing way:

f ri =

[θsSw]L+1i − [θsSw]Li

vi∆t −

∑j∈ηi


L+1j − hL+1

i )−QL+1i (3.89)

where r represents the iteration level. Application of the Newton method to Equation 3.89produces the following matrix equation:

F rij∆ψr+1j = −f ri (3.90)

which can be solved with the same preconditioned iterative solver used for linear matrixequations, since the Jacobian matrix Fij is linear.

In Equation 3.90, the Jacobian matrix, F rij , is defined as:

F rij = ∂f ri∂ψrj

(3.91)

and vector f ri represents the residual of the discretized equation. The iteration process iscarried out repeatedly until the change in the pressure head, ∆ψr+1

j , or the residual of theequation, f ri , becomes less than a specified tolerance at all the nodes. It is important thatthe evolution of the residual, f ri , be monitored during iteration to ensure proper convergence.

Full Newton iteration can be computationally expensive mainly because of the need toevaluate the Jacobian matrix. It is therefore highly desirable to implement a scheme capableof evaluating it in an efficient manner. One option is to evaluate the Jacobian numerically[Forsyth and Simpson, 1991; Forsyth and Kropinski, 1997]. Term by term evaluation of theJacobian can be represented by [Forsyth and Kropinski, 1997]:

∂f ri∂ψri

≈ f ri (ψri + ε)− f ri (ψri )ε

(3.92)

where ε represents a small numerical shift in the pressure head value.

Obviously, from Equation 3.92, numerical differentiation requires more than one functionevaluation for each term; however, it can be shown that not all terms in the Jacobian willrequire two function evaluations. The form of the discretized Equation 3.89 also makes itintuitively easy to evaluate the Jacobian numerically. This is because the fluid flow termsappearing in the summation in Equation 3.89 depend only on nodes i and j. Forsyth andSimpson [1991] and Forsyth and Kropinski [1997] give a detailed procedure for the Jacobianevaluation and it is reproduced below for completeness.

The diagonal term of the Jacobian for node i can be determined from Equations 3.89 and3.91 as:

∂f ri∂ψri

= ∂

∂ψri

[θsSw]ri − [θsSw]Li

vi∆t −

∑J∈ηi

∂

∂ψri(λ)r(ij+1/2)γij(hrj − hri ) −

∂Qri∂ψri

(3.93)

The entries in column i of the Jacobian will be, excluding the diagonal:

Fji =∂f rj∂ψri

(3.94)


where j ∈ ηi. As stated previously, the only term in f rj depending on ψri will be the onerepresenting flow from node i to node j. Therefore:

∂f rj∂ψri

= − ∂

∂ψri(λ)r(ji+1/2)γji(hri − hrj) (3.95)

Because of local conservation of mass, the fluid flow term between i and j appearing in theequation for node i will be similar to the one appearing in equation for node j. Thereforewe have:

λij = λji

γij = γji (3.96)Using Equation 3.96, 3.95 becomes:

∂f rj∂ψri

= ∂

∂ψri(λ)r(ij+1/2)γij(hrj − hri ) (3.97)

The right hand side of Equation 3.97 is also found in the summation appearing in Equa-tion 3.93. The expression for the diagonal term (Equation 3.93) can therefore be expressedas:

∂f ri∂ψri

= ∂

∂ψri

[θsSw]ri − [θsSw]Li

vi∆t −

∑j∈ηi

∂f rj∂ψri

− ∂Qri∂ψri

(3.98)

which shows that the evaluation of the diagonal term for node i incorporates all the termsappearing in column i of the Jacobian. The Jacobian can therefore be constructed byonly evaluating the diagonal terms and subsequently filling in the off-diagonal terms in acolumn-wise fashion.

Forsyth and Simpson [1991] and Forsyth and Kropinski [1997] show that for n unknowns andwith the summation in Equation 3.89 extending from unity to ηi, the building of the Jacobianrequires n(2 + 2ηi) function evaluations. The Picard iteration scheme, on the other hand,requires at least n(1 + ηi) function evaluations to compute the residual, which is seen to beonly a factor of two less than the more robust Newton method. Numerical differentiation isalso attractive because it allows easy use of tabular data to represent arbitrary constitutiverelations should analytical expressions be unavailable.

3.13.3 Primary Variable Substitution

Forsyth et. al. [1995] discuss the use of a saturation based form of Equation 2.1 which hasgood convergence properties in terms of nonlinear iterations when compared with a pressurebased method. Because this method cannot be used in the saturated zone they define a newvariable which is essentially the saturation in the unsaturated zone and the pressure head inthe saturated zone.

They use full Newton iteration to solve the discrete equation everywhere and, in the caseof a constant air phase pressure approximation, which applies here, they simply use adifferent primary variable in different regions. Note that primary variables are regarded asindependent when constructing the Jacobian.


The primary variable at a given node may be switched after every Newton iteration usingthe following method:

IFSi > tolf Use ψi as primary variable at node i

ELSE IFSi < tolb Use Si as primary variable at node i

ELSEDo not change primary variable at node i

ENDIF

(3.99)

with the requirements that:tolf < 1 (3.100)

and:tolf 6= tolb (3.101)

3.13.4 Time Stepping

A variable time-stepping procedure similar to the one outlined by Forsyth and Sammon[1986] and Forsyth and Kropinski [1997] has been incorporated in the solution procedure. Thetime step is defined according to the rate of change of the solution unknown. For flow, therate of change of hydraulic head, pressure head or saturation can be chosen. For transport,the rate of change of concentration or temperature is used for variable time-stepping.

Let’s assume that X represent one of the unknowns mentioned above. After obtaining thesolution at time level L, the next time-step is selected according to:

∆tL+1 = Xmax

max | XL+1i −XL

i |∆tL (3.102)

where Xmax is a desired maximum change in the unknown during a single time-step, definedby the user.

This implementation of variable time-stepping allows the use of increasingly larger timesteps if the dependent variable does not experience drastic changes. It is also recognizedthat the number of Newton iterations taken to achieve convergence is a good indicator ofthe suitability of the current time-step size. If the number of iterations exceeds a specifiedmaximum, ITmax, at time level L+ 1 in which the time step is currently ∆t, the solution isrestarted at time level L and the time-step is reduced, typically by a factor of two. Overall,the procedure can lead to a very significant reduction in computational effort, especiallywhen the solution is desired only at a few widely-spaced target times (see Therrien andSudicky [1996]).

3.13.5 Mass Balance

Mass balance calculation are done by HydroGeoSphere after solving for fluid flow andsolute transport. The mass balance calculation gives information to the user on fluid flow


and solute exchange through external boundaries. It also provides information on theaccuracy of the numerical solution since the discretized flow and transport equations (forexample, Equations 3.18 and 3.65 for the porous medium) are in a mass conservative formand are actually expressions for mass balance for the volume associated to a node in thegrid. These equations are assembled into a global matrix whose solution is obtained byusing iterative methods. Therefore, if the iterative process for solving the matrix convergeswithin a prescribed tolerance, mass balance for each equation should be ensured to the levelof precision of the convergence tolerance.

Mass balance for fluid flow and solute transport is performed by first computing the totalmass change in the domain for a given time step (for steady-state simulations, there is noglobal mass change in the domain). The mass entering or leaving the domain through theboundaries or through internal sources and sinks is then computed and is compared to thetotal mass change, providing the mass balance check for the simulation. The procedure usedto compute the mass entering or leaving the domain is similar to that described in Huyakornand Pinder [1983] and involves reassembling the discretized equation for the first-type nodes.

3.13.6 Solution Procedures

The solution methodology for 2-D areal surface flow is embedded into the time loopingof the solution methodology for the subsurface calculations of HydroGeoSphere (i.e., ateach iteration, assembly of the matrix of flow equations for the subsurface is followed byassembly of its surface flow equations). The entire implicit system of matrix equations isthen solved at each nonlinear iteration until convergence before proceeding to the next timestep. Adaptive time-stepping, Newton-Raphson linearization, and under-relaxation formulasused for the solution are discussed in Chapter 3 among the sections that document solutionto the subsurface equations.

Chapter 4

Verification Examples

In order to verify the numerical techniques and demonstrate applicability of HydroGeo-Sphere via simulation examples, verification is performed by comparison with availableanalytical solutions, published numerical solutions of simulators with some equivalent features,and field or experimental applications.

4.1 Subsurface Flow

4.1.1 Level 1: Drawdown in a Theis Aquifer

In order to verify the accuracy of HydroGeoSphere in simulating drawdown due topumping in a Theis aquifer, it was compared with an exact analytical solution. Theexample is taken from Freeze and Cherry, [1979], to which the reader is referred for detailedinformation regarding the Theis solution. The input parameters for the analytical solutionare shown in the Table 4.1.

In the numerical model, a circular grid of 28596 prism elements which was 10000 m indiameter and 1 m thick was generated. The pumping well was simulated with a singlevertical line element which was located at the center of the grid. The discharge was specifiedat the lowermost node in the well. A prescribed head boundary condition was maintained atthe outer edge of the domain.

Table 4.1: Parameter Values for Simulation of Theis Problem.Parameter Value UnitPumping rate 4.0× 10−3 m s−1

Hydraulic conductivity 0.0023 m s−1

Aquifer thickness 1.0 mAquifer storativity 7.5× 10−4

Radial distance to observation point 55 m

84

CHAPTER 4. VERIFICATION EXAMPLES 85

Time(s)

Dra

wdo

wn(

m)

101 102 103 104 105 106 107 10810-3

10-2

10-1

100

101

Theis Well Function

101 102 103 104 105 106 107 10810-3

10-2

10-1

100

101

HydroGeoSphere

Figure 4.1: Results for Pumping in a Theis Aquifer.

For comparison, the same problem was run using the axisymmetric option with a gradedmesh consisting of 33 block elements, ranging from 0.01 m near the well to 1000 m near theouter boundary.

Drawdown versus time data for a node located 55 m from the pumping well is shown inFigure 4.1.

The results from both the full 3-D grid and axisymmetric grid are very close to those obtainedfrom the analytical solution. Both solutions drop below the Theis solution at late timedue to the influence of the constant head boundary condition. Although both approachesgive essentially identical results, the axisymmetric option results in a 2 order-of-magnitudedecrease in CPU time.

4.1.2 Level 2: Unsaturated Flow Through a Column

This verification example consists of one-dimensional transient infiltration in an unsaturatedvertical column. The specifications of the problem are taken from Huyakorn et al. [1986],Example 4. The physical system is 200 cm long in the vertical (z) direction, with the topface corresponding to the soil surface and the bottom face corresponding to the water table.The column has dimensions of 50 cm in each of the horizontal (x and y) directions. Initially,the pressure head at the water table is zero, it is -90 cm at the soil surface and equals -97cm in the remainder of the domain. Infiltration at the rate of 5 cm d−1 is then applied for


a period of 10 days. The saturated hydraulic conductivity of the soil is 10 cm d−1 and itsporosity is 0.45. The constitutive relationships for the soil are given by:

krw = (Sw − Swr)(1− Swr)

and:(ψ − ψa)

(−100− ψa)= (1− Sw)

(1− Swr)where the residual saturation, Swr, is 0.333 and the air entry pressure , ψa, is 0.0 cm.Substituting these values in the equations given above yields simple linear relationshipswhich can be input to the model in tabular form. The input values of water saturationversus pressure head are shown in Table 4.2. The input values of relative permeability versus

Table 4.2: Water Saturation Versus Pressure Head Relationship for the Unsaturated ColumnExample.

ψ(cm) Sw-0.01 .3330.0 1.0

water saturation are shown in Table 4.3.:

Table 4.3: Relative Permeability Versus Water Saturation Relationship for the UnsaturatedColumn Example.

Sw Krw

.333 0.01.0 1.0

The column is discretized in three dimensions with 2 nodes in each of the x- and y-directionsand 41 nodes in the z-direction. The mesh thus consists of a total of 164 nodes and 40elements. The time steps are identical to Huyakorn et al. [1986] with an initial value equalto 0.1 days, which is increased by a factor of 1.2 until a maximum of 1.0 days is attained.The tolerance on pressure head for the Newton-Raphson iteration is set to 0.01 cm.

Figure 4.2 shows pressure head profiles at 4 different times during the infiltration event fromHuyakorn et al. [1986]. For comparison, results from HydroGeoSphere are also presented.It can be seen that the results are almost identical.

4.1.3 Level 2: Very Dry Initial Conditions

This verification problem is taken from Forsyth et. al.[1995], Example 2, which was developedto compare the performance of numerical simulators for very dry initial conditions. Thecomputational domain is shown in Figure 4.3. All boundaries are no flow except for thezone of infiltration at the top left corner. Table 4.4 provides the material properties for the


Table 4.4: Material Properties for the Simulation of Forsyth et al. [1995], Example 2.Parameter Value UnitSoil Zone 1

porosity, n 0.3680permeability, k 9.3× 10−12 m2

Van Genuchten parameter, α 0.0334 cm−1

Van Genuchten parameter, β 1.982residual saturation, Sr 0.2771

Soil Zone 2porosity, n 0.3510permeability, k 5.55× 10−12 m2










Pressure head (cm)

Hei

ghta

bove

wat

erta

ble

(cm

)

-100 -80 -60 -40 -20 00

50

100

150

200

0.1 days1 day2.6 days10 days

Huyakorn et. al. [1986]

-100 -80 -60 -40 -20 00

50

100

150

200

0.1000000001.092992002.6958682110.9496603

HydroGeoSphere

Figure 4.2: Pressure Head Profiles for the Unsaturated Flow Verification Example.

4 soil zones. They report using a 90 × 21 finite volume grid to discretize the domain, butthe exact grid coordinates were unavailable. The initial pressure head was set to -734 cm,and water infiltration occurred for 30 days.

Figure 4.4 compares saturation contours between HydroGeoSphere using upstream weight-ing and Forsyth’s one phase, central weighting case. The saturation front produced byHydroGeoSphere is considerably sharper than that shown by Forsyth.

4.1.4 Level 2: Drainage of a Fractured Tuff Column

An example is now presented to verify the variably-saturated flow solution in discretely-fractured porous media. Because the variably-saturated flow equation is nonlinear andanalytical solutions are at best approximate, the numerical formulation was verified bycomparison to the numerical solution presented by Wang and Narasimhan [1985] for anexample problem which involves the vertical drainage of a three-dimensional fractured tuffcolumn.

Analytical expressions describing the fracture relative permeability, kr, saturation, S andeffective fracture-matrix flow area, σ, as presented by Wang and Narasimhan [1985] havethe following forms:

kr(ψ) = 16(4 + βbc)

[24− exp(−βbs)(24 + 24βbs + 12β2b2s + 4β3b3s + β4b4s)]


Figure 4.3: Schematic for Very Dry Initial Conditions Problem.


Horizontal Distance (cm)

Ele

vatio

n(c

m)

0 200 400 600 8000

100

200

300

400

500

600 Forsyth et. al. [1995]

HydroGeoSphere

0.6

0.5

0.4

0 200 400 600 8000

100

200

300

400

500

600

Figure 4.4: Results For Very Dry Initial Conditions Problem

+ βbc[6− exp(−βbs)(6 + 6βbs + 3β2b2s + β3b3s)] (4.1)

S(ψ) = 12 + βbc

[2− exp(−βbs)(2 + 2βbs + β2b2s)] + βbc[1− exp(−βbs)(1 + βbs)] (4.2)

σ(ψ) = 1− exp(−βbc − βbs)(1 + βbc + βbs) (4.3)

where the β values are parameters determined from fracture spacing. The variable bc is thecontact cutoff aperture for the fracture and can be determined by the root of the equation:

1− exp(−βbc)(1 + βbc) = ω (4.4)

ω being the fraction of contact area for a fracture.

The variable bs represents the saturation cutoff aperture and is given by:

bs = −2χ cos Θρgψ

(4.5)

where g is the acceleration due to gravity, ρ is the density of water, χ is the surface tension,Θ represents the angle between the solid and liquid surface and ψ is the pressure head.


Table 4.5: Parameter Values Used for Wang and Narasimhan [1985] Relationships.Parameter Value UnitFluid density ρ 1000 kg m−3

Acceleration due to gravity g 9.80665 m s−2

Surface tension γ 0.07183 kg s−2

Solid/liquid surface angle Θ 0.0Fracture contact area ω 12 %Horizontal fracture contact cutoff aperture bc(H) 0.074 mmVertical fracture contact cutoff aperture bc(V ) 0.057 mmβH 0.804× 104 m−1

βV 1.04× 104 m−1

The values of the above variables that are used in this simulator are based on the valuespresented in the Wang and Narasimhan [1985] study, and are show in Table 4.5. Theyexamined flow in a fractured porous tuff unit, the Topopah Spring Member at YuccaMountain, and developed a theory for computing the unsaturated flow properties of fractureswhich was then applied to this rock unit. Based on observations, they obtained values for allthe parameters needed to describe unsaturated flow in the fractured tuff unit. They used anintrinsic permeability of 1.02× 10−11m2 for the fractures. Note that they identified two setsof fractures, vertical fractures, for which the subscript V is used, and horizontal fracturesdenoted by subscript H.

The theoretical expressions developed by Wang and Narasimhan [1985] to describe thesaturation, relative permeability and effective fracture-matrix flow area for the fractures, asfunctions of the fluid pressure, were implemented in this model and used for the simulation.The comparison was made for the case where the phase-separation constriction factor, whichwas used by Wang and Narasimhan [1985] to represent the effects of the air phase on theflow of water, was neglected.

Figure 4.5 illustrates the geometry of the physical system. The porous tuff matrix containsthree fracture sets, two sets are vertical with a constant fracture aperture equal to 240 µmand one set is horizontal, with a fracture aperture equal to 310 µm. It should be noted thatthe fractures have not been drawn to scale in Figure 4.5. The fractures partition the matrixinto blocks, with each block having dimensions equal to 0.22 m × 0.22 m × 0.48 m. A totalof 27 such blocks are represented in Figure 4.5. The saturated hydraulic conductivity of thetuff matrix is 3.2× 10−8 cm s−1, its porosity is 0.09 and its specific storage equals 1× 10−6

m−1. The constitutive relations describing matrix saturation and relative permeability arerepresented by the Van Genuchten relations (Equations 2.7 and 2.8), with Swr = 9.6× 10−4,α = 7.027 × 10−3 m−1, m = 0.45 and n = 1.818. The specific storage of the fractures isequal to 4.4× 10−6 m−1.

Due to the symmetry of the system, Wang and Narasimhan [1985] only considered onevertical column bounded by four vertical fractures. For this comparison, only one quarter ofthis column, i.e. total dimensions equal to 0.11 m × 0.11 m × 1.44 m, need be discretized,taking advantage of the horizontal symmetry of the drainage process. The column was


Figure 4.5: Verification Example Involving Fractured Porous Tuff, from Wang and Narasim-han, [1985].


Time (days)

Pre

ssur

e(m

)

10-4 10-2 100 102 104 106-120

-100

-80

-60

-40

-20

0

A,BDC

Wang-Narasimhan

10-4 10-2 100 102 104 106-120

-100

-80

-60

-40

-20

0

ABCD

HydroGeoSphere

Figure 4.6: Pressure Drop at Selected Points During the Drainage of a Fractured PorousTuff.

discretized using 7 nodes in each of the horizontal directions and 31 nodes in the verticaldirection, for a total of 1519 nodes (1080 three-dimensional elements). The nodal spacingused was identical to that reported by Wang and Narasimhan [1985]. Each vertical fracturewas represented by 180 two-dimensional elements and 36 elements were used to discretizeeach horizontal fracture.

The column was initially saturated, the fluid was static and its potential was everywherezero. Drainage was performed by applying a suction equal to −112.0 m at the bottom ofthe column, all other boundaries being impermeable. This suction caused the fractures andthe matrix to desaturate with time. Time stepping control was used to move the solutionthrough time. A maximum change in pressure head of 1.0 m for each time step was used inconjunction with Equation 3.102. The total CPU time for the flow simulation was 6 minuteson the IBM RS/6000 Model 590 and a total of 511 variable time steps were necessary toreach the final simulation time of 105 years. It should be noted that convergence of theNewton procedure occurred typically after the first iteration for most time steps.

A comparison of the results obtained with this model and those of Wang and Narasimhan[1985] is presented in Figure 4.6, which shows the change in fluid pressure with time for fourdifferent locations in the column. Location A represents the middle of the porous block,location B is the middle of a vertical fracture bounding the matrix block and points C andD are in the middle of the horizontal fractures. It can be seen from Figure 4.6 that thereis a very good agreement between the results obtained and those reported by Wang andNarasimhan [1985]. The pressure head is seen to decrease gradually at early times for allobservation points. The decrease is more rapid at point D, which is closer to the drainage


Table 4.6: Parameter Values for Simulation of 1-D Hydromechanical Coupling Problem.Parameter Value UnitHydraulic conductivity 1× 10−3 m yr−1

Specific storage 1× 10−6 m−1

Loading efficiency 1.0

boundary. The pressure at points A and B is identical, revealing that the drainage processfor this case is mainly influenced by the porous matrix when unsaturated conditions prevail.

The drainage simulation was repeated using both the finite element and the finite differencerepresentations. Both representations produced identical results, although the finite differencerepresentation required one quarter of the CPU time compared to the finite element scheme.

4.1.5 Level 1: 1-D Hydromechanical Coupling

This verification example consists of the one-dimensional transient head response to uniformloading on the top of a saturated vertical column. The specifications of the problem aretaken from Lemieux [2006], who developed an exact analytical solution for a column ofsemi-infinite length, which is subjected to loading by a mass M , added at constant intervalssuch that dM/dt is constant and where the top of the column is drained and the base is ano-flow boundary condition.

HydroGeoSphere was used to model this case and was compared to the analytical solution.A domain of 10,000 m length was used. A load of 0.3 m/yr was applied for a period of10,000 years. Drainage at the top of the column is achieved by specifying a head of 0.0 m atthe top of the column. The initial head along the length of the column was set to 0 m. Theproperties of the rock mass are described in Table 4.6.

Figure 4.7 shows the hydraulic head versus time at different depths in the column fromLemieux [2006]. Also shown is the head response from HydroGeoSphere at a depth of500 m below the top of the column. it can be seen that the numerical solution preciselycorresponds to the analytical solution.

4.1.6 Level 1: 1-D Hydromechanical Coupling with Externally ComputedStresses

This example is used to demonstrate that HydroGeoSphere could be used in conjunctionwith other hydromechanical models.

In general, because HydroGeoSphere cannot generate equilibrated hydromechanical totalstresses for use in Equation 2.21, they must be generated by other models and used asinput to HydroGeoSphere. Theoretically, time-dependent, average, element-specific totalstresses are required.

As shown in Figure 4.8, a column of soil of height 1,000 metres supporting a load Φzz is


Figure 4.7: Results for 1D Hydromechanical Coupling Example.


Table 4.7: Parameter Values for Simulation of 1-D Hydromechanical Coupling Problem.Parameter Value UnitHydraulic conductivity 0.1 m d−1

Specific storage 0.0018 m−1

Loading efficiency 1.0

confined laterally in a rigid sheath so that no lateral expansion can occur. It is assumedthat no water can escape laterally or through the bottom while it is free to escape at theupper surface. The height of 1000 m is assumed to reflect the magnitude of depth thatHydroGeoSphere is likely to be applied. Material properties for the soil are given in Table4.7.

A vertical stress of 0.5 MPa was applied to the surface, which causes the water pressurehead to rise by 50.9 metres at the onset.

For cases of purely vertical strain such as we have here, Guvanesan [2007] presents a methodfor computing the average hydromechanical stresses at any time from water pressuresobtained from an analytical solution developed by Biot [1941]. Details of the mathematicaltheory and procedures are not given here, but instead the reader is referred to Guvanesan[2007].

The analytical solution of Biot [1941] is based on the assumption that the vertical load isinstantaneously applied and remains constant thereafter. In applying HydroGeoSphere,it was assumed that the load was increased in a linear fashion from 0 to 50.9 m in 0.1 days.

Average hydromechanical stresses versus time (expressed as water pressure in m) computedby Guvanesan [2007] at various elevations are plotted in Figure 4.9. These stresses wereused as input to HydroGeoSphere

Figure 4.10 shows the hydraulic head versus time at different elevations in the column fromBiot [1941] and HydroGeoSphere. The agreement between HydroGeoSphere and theanalytical solution is an indication that HydroGeoSphere can be interfaced with otherhydromechanical models.

4.2 Surface Flow

4.2.1 Level 1: 1-D Surface Flow Study of Govindaraju et al., [1988a and1988b]

The surface water flow capabilities of HydroGeoSphere are verified against analytical andnumerical solutions of diffusive wave and dynamic wave equations, presented by Govindarajuet al. [1988a and 1988b]. The problem considered involves one-dimensional surface flow alonga plane of one unit width (Figure 4.11). The authors presented numerical and analyticalsolutions for the different waves under a wide range of flow conditions ranging from highlysubcritical flow (Froude number Fo = uo/

√gdo < 0.5) to supercritical flow (Fo = 1.5) and at


Figure 4.8: Schematic for 1D Hydromechanical Coupling with External Stresses Example.


Figure 4.9: Mean Normal Stress Versus Time at Various Elevations.


Figure 4.10: Results for 1D Hydromechanical Coupling with External Stresses Example.


Figure 4.11: Schematic Description of the Govindaraju et al. [1988a and 1988b] Problem.


Table 4.8: Parameter Values for Simulation of a 1-D Flow Example from Govindaraju et al.[1988a and 1988b].

Parameter Value UnitFroude number Fo 0.5Kinematic wave number K 10 m s−1

Slope So 0.01Slope length L 100 mUniform recharge io 0.0040 m s−1

Uniform velocity uo 0.9905 m s−1

Uniform depth do 0.4000 mManning’s roughness coefficient n 0.0548 s m−1/3

different kinematic wave numbers K(= SoL/doF2o ) ranging from 3 to 50 (K > 20 indicates

a kinematic wave), where uo is the uniform velocity and do is the uniform depth at thedownstream end, So is the bed slope, and L is the slope length.

We chose a single case for comparison, using the parameters shown in Table 4.8.

A zero-depth gradient boundary condition (see Equation 3.57) was applied at the downstreamend of the system.

Modeling Approach and Results

The slope was discretized into 100 columns and 1 row of elements with dimensions 1m× 1m each. Only surface flow resulting from recharge was simulated and no interactionwith ground water was considered. Adaptive time stepping was provided in the simulationswith an initial time-step size of 1s and a maximum time-step size of 2.5s. Newton iterationconsiderations include a maximum of 20 iterations an absolute convergence tolerance of10−4. Figure 4.12 shows the discharge hydrograph at the downstream end of the slope. Thehydrograph was normalized by dividing the discharge by the normal discharge Qo, in thiscase 0.3962 m3 s−1, and time by to(to = L/uo).

The results obtained using HydroGeoSphere are in good agreement with both the diffusionwave solution of Govindaraju et al. [1988a and 1988b], where Picard and Newton iterativeapproaches provide almost identical solutions, and the Modflow-Surfact solution. However,due to the limitations of the diffusive wave approximation, the solution deviates from thedynamic wave (i.e., the full Saint Venant solution) for the small value of the kinematic wavenumber K used in this example.

4.2.2 Level 2: Conjunctive Surface-Subsurface Flow Study of Smith andWoolhiser, [1971]

Smith and Woolhiser [1971] present an experimental study of combined surface and subsurfaceflow which may be used to validate the surface and subsurface flow routines, and theirinteractions, in HydroGeoSphere. The experiments consisted of providing rainfall of 25


Normalized Time

Nor

mal

ized

Dis

char

ge

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Diffusion wave approximationFull St. VenantModflow-Surfact

Normalized Time

Nor

mal

ized

Dis

char

ge

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

HydroGeoSphere

Figure 4.12: Comparison of Normalized Rising Hydrographs for Saint Venant Equations,the Diffusion Wave Approximation [Govindaraju et al., 1988a] and MSVMS for Fo = 0.5and K = 10.


cm hour−1 for 15 minutes over a soil flume 1,220 cm long, 5.1 cm wide and 122 cm deep.The flume was inclined along its length at a slope of 0.01, and the moisture movement intothe soil, and downstream discharge were measured. The experimental setup is schematicallydepicted in Figure 4.13. Several conjunctive models of surface and subsurface flow haveselected this experiment as a test case. Smith and Woolhiser [1971] present a model usingthe kinematic wave approximation for surface flow. Akan and Yen [1981] and Singh andBhallamudi [1998] solve the full dynamic wave equation for surface flow. Govindaraju andKavvas [1991] solve the diffuse wave equation for surface flow for this case. All above modelssolve the Richards equation for flow in the subsurface, and use flux coupling between thesurface and subsurface equations.

The case considered for simulation here, involves a dry soil. The soil, a Poudre fine sand,was placed in the flume in three layers of thickness 7.65 cm, 22.95 cm, and 76.1 cm from topto bottom (denoted as layers 1, 2, and 3, respectively). Hydraulic property curves providedby Smith and Woolhiser [1971] for the 3 soil layers were fit to the Van Genuchten functionsusing a regression code that adjusted the saturated moisture content and saturated hydraulicconductivity values. Figure 4.14 shows the observed moisture retention curve for soil layer 1,as provided by Smith and Woolhiser [1971], and the fitted Van Genuchten functions, andindicates that the Van Genuchten parameters used for the simulation provide a good fitto the experimental data in the range of suction valid for this study. Figure 4.15 showsthe observed and fitted relative hydraulic conductivity vs. moisture content curves for soillayer 1. The fitted Van Genuchten functions for the other two soil layers, not shown here,also had very good fits. Singh and Bhallamudi [1998] provide details of the Brooks-Coreyrelations fitted to the data of Smith and Woolhiser [1971]. Table 4.9 provides a summaryof the fitted Van Genuchten parameter values of the simulation for all 3 soil layers. Akanand Yen [1981] and Singh and Bhallamudi [1998] provide the Darcy-Weisbach friction factorused for the simulation as f = CL/Re where the coefficient CL is 92 for this laminar flowproblem, and Re is the Reynolds number defined as Re = q/µ, where q is the discharge perunit width and µ is the kinematic viscosity of the liquid. The liquid used in the experimentswas a light oil with a kinematic viscosity of 1.94× 10−6 m2 s−1. The average discharge forthe experiments was approximately 1× 10−5 m3 s−1 for a flume width of 0.051 m, givingq = 10−5/0.051 = 1.96 × 10−4 m2 s−1 and thus Re = 101.07, for an average number todetermine the friction factor f as 92/101.07 = 0.91. The Darcy-Weisbach equation, Chezy’sequation and Manning’s equation are related as Cc = d1/6/n =

√8g/f where d is the depth

and g is gravity, thus giving the Chezy constant for this problem as Cc =√

8g/f = 5569.1cm1/2 minute−1 and the Manning constant as n = d1/6/C = 0.000180d1/6 minutes cm−1/3.The depth of flow for this problem is less than 1 cm - assuming it to be 0.15 cm givesd1/6=0.6873 (the one-sixth power brings it all closer to unity). Thus, Manning’s n for thisproblem is 0.0001228 minutes cm−1/3.


The two-dimensional domain was discretized into 100 columns, 1 row and 40 layers ofelements. The top 5 model layers represent soil layer 1, the next 15 model layers representsoil layer 2 and the remaining 20 model layers represent soil layer 3. The length of eachelement was 12.2 cm with a width of 5.1 cm and a thickness of 1.53 cm in soil layers 1


Figure 4.13: Experimental Setup of the Smith and Woolhiser [1971] Study.


Suction pressure (cm)

Moi

stu

reC

ont

ent

Hyd

rau

licC

on

duc

tivity

(cm

/min

)

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0

0.05

0.1

0.15

0.2

0.25

Observed moisture contentFitted moisture contentObserved Hydraulic ConductivityFitted Hydraulic Conductivity

Figure 4.14: Moisture Retention Curve for Soil Layer 1.

Moisture Content

Rel

ativ

eH

ydra

ulic

Con

duc

tivity

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

ObservedFitted

Figure 4.15: Hydraulic Conductivity versus Soil Moisture for Soil Layer 1.


Table 4.9: Parameter Values for Simulation of the Smith and Woolhiser [1971] Experiment.Parameter Value UnitSoil layer 1 †

porosity, n 0.3946saturated conductivity, Ks 0.184 cm minute−1



Soil layer 2porosity, n 0.4387saturated conductivity, Ks 0.1452 cm minute−1



Soil layer 3porosity, n 0.4764saturated conductivity, Ks 0.1296 cm minute−1



Darcy-Weisbach friction, fr 0.91Equivalent Chezy coefficient, Cc 5569.1 cm1/2 minute−1

Equivalent Manning coefficient, n 0.000122 minute cm−1/3

Rainfall intensity, i 0.417 cm minute−1

Channel slope 0.01Channel length 1220 cmChannel width 5.1 cmgravitational acceleration 3.532× 106 cm minute2

† The soil parameters provide the fit to experimental data of Smith andWoolhiser [1971], as shown in Figures 4.14 and 4.15 for soil layer 1.


Saturation

Dep

th(c

m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

10

20

30

40

50

Singh and Bhallamudi (1998)Modflow-Surfact

0.100000000 minutes6.00000000 minutes8.50000000 minutes10.3000000 minutes15.0000000 minutes

HydroGeoSphere

Figure 4.16: Soil Saturation Profile at 550 cm from Upstream End for Simulation of theSmith and Woolhiser [1971] Study.

and 2, and a thickness of 3.805 cm in soil layer 3. The entire rectangular domain is tiltedalong its length to provide a slope of 0.01 to the model domain. The dual node approach,with a coupling length of 1.35 m and a rill storage height of 0.01 m was used to couplethe surface and subsurface flow systems. Rainfall at a rate of 0.416667 cm minute−1 issupplied for a duration of 15 minutes followed by 5 minutes with zero recharge. Adaptivetime-stepping uses an initial time-step size of 0.01 minutes and a maximum time-step size of1 minute. Newton iteration considerations include a maximum of 18 iterations an absoluteconvergence tolerance of 10−3 cm. A critical-depth gradient boundary condition is appliedat the downstream end of the surface flow nodes, while the lateral and bottom boundariesof the subsurface are provided no flow conditions.

The soil is initially dry at an approximate saturation of 0.2 as shown by Smith and Woolhiser[1971], which is converted to the appropriate head value using the Van Genuchten relationfor the appropriate soil layer. The initial water depth for the surface flow nodes was set to 1×10−4 m to represent dry starting conditions at the surface. Simulations were performedusing the Newton-Raphson scheme, as well as using Manning’s equations to represent surfaceflow conditions.

Figure 4.16 shows the saturation profiles within the soil at a distance of 550 cm from theupstream end at different times, and show that infiltration causes the saturation front toadvance within the soil. The infiltration front predicted by HydroGeoSphere compares


Time(minutes)

Dis

char

ge(c

m3 /m

inut

e)

6 8 10 12 14 16 18

-1200

-1000

-800

-600

-400

-200

0

Smith and Woolhiser, 1971Singh and Bhallamudi, 1998Modflow-Surfact, ManningModflow-Surfact, Darcy-Weisbach

6 8 10 12 14 16 18

-1200

-1000

-800

-600

-400

-200

0

HydroGeoSphere, Manning, Dual node scheme

Figure 4.17: Outflow Hydrograph for Simulation of the Smith and Woolhiser [1971] Study.

fairly well with other simulation attempts [Singh and Bhallamudi 1998; Smith and Woolhiser,1971] and with available experimental data.

Figure 4.17 shows the outflow hydrograph at the downstream end of the surface flow nodes.This compares well with experimental data, and with other simulation attempts using thekinematic wave equation [Smith and Woolhiser, 1971] or the dynamic wave equation [Singhand Bhallamudi, 1998].

Figure 4.18 shows the surface water depth profiles at 8.3, 15 and 16 minutes of simulationrespectively. The depth is noted to increase in time up till the end of the recharge period,and rapidly dissipates thereafter.

Figure 4.19 shows key components of the fluid balance for the simulation. The data areexpressed as flow rates in centimetres cubed per minute, so, for example, the rainfall rateshows up as a uniform value of 2592 cm3 minute−1 until a time of 15 minutes, when it dropsto zero. Initially, most of the rainfall is taken up by the subsurface, and accumulation in thesurface flow domain is negligible before a time of about 3 or 4 minutes. Accumulation in thesurface flow domain peaks at about 7 minutes and then declines as water begins dischargingat the critical depth boundary. Fluid balance errors, which in this case are the differencebetween the rainfall plus the critical depth accumulation rates and the subsurface plus thesurface domain accumulation rates, were less than a fraction of a percent throughout thesimulation.


Distance along channel (cm)

Wat

erde

pth

(cm

)

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

8.5000 minutes15.000 minutes16.000 minutes

Figure 4.18: Surface Water Depth Profiles at Different Times for the Simulation of the Smithand Woolhiser [1971] Study.


Time (minutes)

Flo

wra

te(c

m3 /m

inut

e)

0 5 10 15

-2000

-1000

0

1000

2000

3000

RainfallCritical depth outflowSubsurface accumulationSurface accumulationError

Figure 4.19: Fluid Balance Results for the Simulation of the Smith and Woolhiser [1971]Study.


4.2.3 Level 2: 2-D Surface Flow Study of diGiammarco et al., [1996]

Two-dimensional areal surface flow is verified using the rainfall-runoff example of di-Giammarco et al. [1996]. VanderKwaak [1999] presents details of the simulation, withresults from various surface water flow codes benchmarked against this problem. The prob-lem involves two-dimensional surface flow from a tilted V-catchment (Figure 4.20) generatedby a 90 minute duration, 3 × 10−6 m s−1 intensity rainfall event. Only one half of thedomain need be simulated due to symmetry, with published outflow discharge halved, toproduce equivalent results. The simulation domain therefore consists of a 1,000m by 800mslope connected to 1,000m length of channel 10m wide. Surface slopes are 0.05 and 0.02,perpendicular to, and parallel to the channel respectively. Manning’s roughness coefficientsof 0.15 and 0.015 are applied to the slopes and channel, respectively. A critical depthboundary condition is applied at the downstream end of the channel.


The domain was discretized into 10 rows and 9 columns, with the last column representingthe 10m wide channel. The remaining grid blocks are 100m × 100m. Only surface flow wassimulated, with rainfall applied at the rate of 3× 10−6 m s−1 for 90 minutes, followed by norainfall for the second stress period of 90 minutes. Adaptive time stepping with an initialtime-step size of 5s, a maximum time-step size of 100s and a time-step incrementing factorof 2.0 is used for the simulation. Newton iteration considerations include a maximum of 20iterations an absolute convergence tolerance of 10−4.

Figure 4.21 compares predicted discharge from HydroGeoSphere with predictions fromseveral other codes. Excellent agreement is noted between all results. Figure 4.22 shows thechannel stage at two points at the outlet: one near the centre of the channel and one near theedge of the channel. The channel stage from Modflow-Surfact, a cell-centred finite differencemodel, is seen to fall within the edge and centre channel stages of HydroGeoSphere. Thestage at the outlet point is noted to increase and dissipate in accordance with the computeddischarge fluxes. Most time-steps converged within two iterations, with negligible massbalance errors throughout the simulation.


4.3.1 Level 1: Groundwater Pumping and Observation Wells

Groundwater pumping perturbs the hydraulic head distribution and creates a cone ofdepression around the pumping well. In confined aquifers, water movement is essentiallyradial or horizontal shown in Figure 4.23 (a). However, unconfined aquifers have vertical flowabove the aquifer shown in Figure 4.23 (b). While there are numerous groundwater analyticrelations between pumping rates and drawdown, the most common method to evaluatesubsurface response is a constant rate pump test in a confined or unconfined aquifer.

For the purpose of testing one-dimensional features, we implemented a simple cylinder-shape


Figure 4.20: Schematic Description of 2-D Surface Water Flow Study of diGiammarco et al.[1996].


Time(seconds)

Dis

char

ge(m

3 /sec

ond)

0 2000 4000 6000 8000 10000

-2.5

-2

-1.5

-1

-0.5

0

CVFEMIFDMSHEModflow-Surfact

0 2000 4000 6000 8000 10000

-2.5

-2

-1.5

-1

-0.5

0

Hydrosphere

Figure 4.21: Outflow Hydrograph for Simulation of 2-D Surface Water Flow Study ofdiGiammarco et al. [1996].


Time(seconds)

Sta

ge(m

)

2000 4000 6000 8000 10000

0.05

0.1

0.15

0.2

0.25

0.3

Modflow-Surfact

2000 4000 6000 8000 10000

0.05

0.1

0.15

0.2

0.25

0.3

HydroGeoSphere,channel centreHydroGeoSphere,channel edge

Figure 4.22: Channel Stage at Outlet for Simulation of 2-D Surface Water Flow Study ofdiGiammarco et al. [1996].


Figure 4.23: Pumping wells create a cone of depression and aquifer properties can beestimated for confined (a) and unconfined (b) aquifers. A 3-D grid system shown in (c) wasused to simulate the pumping and drawdown relation in a cylindrical aquifer.


Figure 4.24: The simulated time-drawdown relations for the common and dual node ap-proaches shown by circles and dashed lines. The results are compared to analytical solutionsrepresented by the solid line for confined aquifers (a) and unconfined aquifer (b) systems.

aquifer domain with a radius of 5 km and thickness of 20 metres shown in Figure 4.23 (c).A fully-penetrating pumping well is placed in the centre of the domain with a screen radiusof 5-cm. The entire porous media domain is set to a homogeneous and isotropic hydraulicconductivity and specific storage of 2.3 × 10−3 m/s and 7.5 × 10−4 m−1. As a means tocompare to the analytical solution, the relative permeability and degree of saturation areassumed to be the same and follow a Gardener exponential function in the unsaturated zone.

Twenty 1-m thick layers with triangular prism elements were used to discretize the subsurfacedomain. The one-dimensional well was discretized with 20 of 1-m long line elements andhad a constant pumping rate of 0.08 m3/s applied to the bottom well node. Two approacheswere implemented to couple the porous medium and well domains.The first method usedthe common node approach that assumes identical hydraulic head values for the well andcontacting porous medium. The second method implemented the dual node approach whichcalculates the fluid exchange between the domains via a first-order leakance relationship.The dual node approach used an interface thickness between two domains of 10−4 m and ahydraulic conductivity equal to the porous medium.

The analytical solutions for the confined (Theis, 1935) and unconfined aquifers (Mathiasand Butler, 2006; Misha and Neuman, 2010, Barlow and Moench, 2011) were comparedto the simulated numerical time-drawdown values for 5 m, 25 m, and 55 m away from thepumping well. The results, shown in Figure 4.24, indicate that the simulated time-drawdowncurves can generally replicate the analytical solutions. However for early time observations,there are notable differences between the two solutions for distant unconfined wells and it isnot clear whether the discrepancy is due to the assumptions used to derive the analyticalsolutions (the thickness of the developed unsaturated zone due to pumping is negligiblecompared to the aquifer thickness) or numerical errors.


Figure 4.25: Steady state results of the numerical and analytical solutions are shown bycircles and solid lines. The hydraulic head, axial flux and radial flux distributions are plottedalong the vertical pumping well axis for confined (a) and unconfined (b) aquifer systems.

The steady-state head, radial flow, and axial flow along the pumping well are illustrated inFigure 4.25. The numerical simulation, shown with circles, used the dual node approach,while the analytical solution with an assumed uniform flow along the saturated portion ofthe well is shown by the solid lines. The results indicate that the simulated hydraulics inand around the pumping well perfectly match the analytic solutions for the confined aquifercase. The unconfined case has some discrepancies above the water table due to the assumedrelations between well conductance and saturation.

4.3.2 Level 1: Subsurface Water Supply Systems

Urban water supply systems typically distribute potable water from the water treatmentfacility to an end user through a network of pipes, tanks, pumps and control valves. Waterdistribution networks always flow from higher head to lower head and the flux and linearhead loss inside of the pipes can be determined with an empirical relationship. Largerdiameter pipes have lower internal velocities and lose less head per unit length than smallerdiameter pipes. Thus, to adjust the tap pressure for the end destination, the pipe diametercan be adjusted to achieve the correct pressure and velocity standards. In order to maintainquality potable water quality throughout the supply system, water mains must maintaina minimum positive pressure to the surroundings so that the outside water cannot enterthe pipe system. On the other hand, pipes may break up if the internal pressure exceeds acertain limit.

For verification of the numerical simulation, a one-dimensional pipe flow problem was imple-mented with the Hazen-Williams equation. The 1-D pipe interacted with the surrounding


Figure 4.26: Simulated (symbols) and analytic (lines) solutions for hydraulic head, andaxial flux distributions along a 100-m long pipe of 10-cm diameter, for a uniform leakage (a:rectangles and dashed lines) and non-uniform leakage (b: circles and solid lines) cases.

porous medium and the results were compared to an analytic solutions. The 100-m long10-cm diameter pipe had a constant pressure head of 30-m at the inlet (x = 100m) and 20-mat the outlet (x = 0m). For illustration and testing purpose, the pipe was assumed to lose75% of the original water flowing into the pipe inlet due to a uniform leakage for the firstcase and non-uniform leakage for the second case. The simulated numerical steady-stateresults predict the same head and flux values as the analytical solutions along the leakingpipes, as shown in Figure 4.26.

4.3.3 Level 2: Subsurface Drain Systems

MacQuarrie and Sudicky (1996) presented an approach to simulate subsurface rectangulardrains by solving a one-dimensional flow equation. The authors demonstrated that theapproach was appropriate and efficient by comparing their numerical solutions to theKirkham analytic solutions (1949) and the two-dimensional numerical solutions provided inFipps et al. (1986). For the comparison MacQuarrie and Sudicky used a three-dimensionaldomain that was 3-m deep (z-axis), 30-m long (x-axis), and 30-m wide (y-axis). They useda saturated hydraulic conductivity of 0.0536 m/hr and adapted the Brooks-Corey soil waterretention relation parameters from Fipps and Skaggs (1991). The simulated porous mediahad a horizontal drain pipe extended from 1-m to 29-m in the y-direction and fixed in the xand z directions (x=15-m, z=2-m). The pipe was specified with a flux boundary conditionapplied at the end of the drain (x=15-m, y=1-m, z=2-m). The steady-state flow case fixedthe water-table was to the top of the ground surface and maintained the drain flux rate at1.7584 m3/hr.

This study changes the rectangular drain used by MacQuarrie and Sudicky (1996) to acircular drain with an effective radius of 0.0108-m as performed by Fipps et al. (1986). The


Figure 4.27: The vertical profiles at x=15m and x=22m are shown in (a) for the simulatedand the analytical solutions by circles and solid line. The simulated water-table profile,shown in (b), is shown after 15 hours of drainage (triangles) and were compared to thenumerical solutions by Fipps et al. (1986) (solid line with circles) and MacQuarrie andSudicky (1996).

new steady state simulated solutions were compaered to the Kirkham analytical solutionsand MacQuarrie and Sudicky numerical solutions. The steady state vertical pressure headprofiles were monitored at two locations, (x=15-m, y=15-m) and (x=22-m, y=15-m), andare shown in Figure 4.27. For the comparison of the transient flow solutions, a no flowcondition was applied at the top ground surface. The transient solution was tested for atotal of 15 hours and the results were compared to the solutions by Fipps et al. (1986) andMacQuarrie and Sudicky (1996) shown in Figure 4.27 (b).

4.3.4 Level 1: Subsurface Sewer Systems and Overland Flow

Sanitary sewer systems, similar to overland flow networks, can be simulated in terms of aseries of one-dimensional flow along a drainage network of streams and rivers. The sewersare designed to collect and transport used water to the waste water treatment facilities undergravity driven flow and the water may flow into or out of the collection system depending onthe conditions of the pipe hydraulics and the surrounding environment. Partially-saturatedflow, as seen in most gravity driven waste water collection systems, is typically describedwith the Manning’s equation.

The implementation of Manning’s formula in one-dimensional structures is tested with a100-m long by 1-m wide rectangular channel. The channel has a constant down streamslope of ∂z

∂s = 0.1 with a constant inflow boundary condition. The inlet surface waterdepth is specified to a zero-m depth while a critical depth boundary condition is appliedat the outlet such that qoutlet = −

√g · d3. Throughout the channel, a constant influx,


Figure 4.28: Simulated water depth and axial flux distributions along a 100-m long 1-mwide rectangular open channel (a). In (b), the simulated depth-flux relation is compared tothe simplified analytic relation when ∂h/∂s ≈ ∂z/∂s shown by circles and solid lines.

qr = 6.02 × 10−2 m3/s/m, is applied resulting in a total stream flow of q(x) = q0 + qr · x.Figure 4.28(a) shows the linear simulated water depth and axial flux distributions results inthe domain. The simulated depth-flux relation is compared to the Manning’s formula where∂z∂s is approximated by bed slope shown in Figure 4.28(b).

4.4 Coupled Surface/Subsurface Flow

4.4.1 Level 3: 3-D Field Scale Study of Abdul [1985]

A field scale simulation is performed with HydroGeoSphere to verify its capabilities for fullythree-dimensional surface/subsurface flow modeling. Experiments conducted at CanadianForces Base Borden, in Ontario, Canada, by Abdul [1985] are selected for this simulation.VanderKwaak [1999] presents details of the site, its characteristics, the experimental setupand the results. Figure 4.29 shows the experimental plot, approximately 80 m × 16 mareally and up to 4 m deep. A man-made stream channel lies approximately 1.2 m belowthe surrounding grassy land. The channel is initially dry prior to the application of theartificial rainfall via irrigation sprinklers. The initial water table lies around 22 cm belowthe streambed with the artificial recharge applied at a rate of 2 cm hour−1 for 50 minutes.Infiltration in upland regions, discharge in lower regions and runoff, all govern the behaviorof the system. Soil properties and roughness coefficients are provided in Table 4.10.


The domain was discretized areally into 1372 nodes and 2651 triangular elements. Vertically,the grid is distorted to conform with the topographic elevation as shown in Figure 4.30.Fifteen layers of elements were used with a fine discretization of 0.1 m near the surface which


Distance (m)

0

20

40

60

80Distance

(m)

0

10

Elevation

(m)

0

2

4

43.5

3

3.5

4.554.543.532.521.51

Figure 4.29: Site Description for Rainfall-Runoff Field Experiment of Abdul [1985] [fromVanderKwaak, 1999].

Table 4.10: Parameter Values for Simulation of the 3-D Field Scale Study of Abdul [1985].Parameter Value Unitporosity, Θ 0.37hydraulic conductivity, K 1× 10−5 m s−1

storage coefficient, Ss, 1.2× 10−7 m−1

Van Genuchten parameter, α 1.9 m−1

Van Genuchten parameter, β 6residual saturation, Sr 0.18Brooks-Corey coefficient, n 3.4Manning coefficient for plot 0.3 s m−1/3

Manning coefficient for channel 0.03 s m−1/3

Initial water table elevation 2.78 m


Distance (m)

0

20

40

60

80Distance

(m)

0

10

Elevation

(m)

0

2

4

Figure 4.30: Three-dimensional View of Topography and Finite element Grid, for Simulationof the Abdul [1985] Rainfall-Runoff Field Experiment.


Figure 4.31: Outflow Hydrograph for Simulation of the Abdul [1985] study.

was enlarged to 1 m near the bottom. The dual node approach was used to simulate surfaceflow. The resulting 3-D finite element mesh is identical to the one used by VanderKwaak[1999]. Recharge is provided at the rate of 5.56 × 10−6 m s−1 for a first stress period of50 minutes, with zero recharge for the second 50 minute stress period. A critical depthboundary condition is applied all around the upper surface of the domain, and the simulationis performed for 100 minutes, with no recharge occurring for the second 50 minute stressperiod. Adaptive time-stepping is used with an initial time-step size of 5s, a maximumtime-step size of 100s, and time-step incrementing and decrementing factor limits of 2.0and 0.5, respectively. Iteration parameters include 100 and 15 inner and outer iterationsrespectively, and a convergence tolerance of 1 × 10−4 m. Figure 4.31 shows the outflowhydrograph to be close to the experimental values, and to the simulation of VanderKwaak[1999]. As noted by VanderKwaak [1999], this system’s outflow hydrograph is extremelysensitive to channel elevations, initial water-table levels, and side-slopes of the plot.

The spatial distribution of water depth within the domain at t=50 minutes is shown inFigure 4.32 for both VanderKwaak [1999] and HydroGeoSphere. These compare well forregions of maximum water depth around the channel but differ elsewhere.

These small differences between the outflow hydrograph and surface water depth betweenthe two simulations may be due to the different approaches used to define the flow couplingterm between the surface and subsurface meshes in the dual node approach. VanderKwaak[1999] used a flow coupling length (in this case 1×10−4 m), height of microtopography (1


Distance

(m)

0

10

20

30

40

50

60

70

80Distance (m)0 5 10 15

Ele

vatio

n(m

)

0

5

0.5

-0.5-1

0

10.50

-0.5-1-1.5-2

log

10

(cm)

(a) (b)

Figure 4.32: Spatial Distribution of Water Depth after 50 Minutes of Field Experiment for(a) VanderKwaak [1999] and (b) HydroGeoSphere.


×10−2 m) and mobile water depth (1×10−4 m) to define the coupling term while Hydro-GeoSphere uses a coupling length of 1 ×10−1 m and a rill storage height of 2 ×10−3 m.Also, VanderKwaak [1999] assigned critical depth boundary conditions to a few nodes at theoutflow end of the channel, while in HydroGeoSphere, the critical depth boundary wasassigned all around the outside of the domain, which would account for the depressions inthe water depth surface observed for HydroGeoSphere in Figure 4.32.

4.4.2 Level 2: 3-D Surface/Subsurface Flow and Evapotranspiration

Evapotranspiration is verified using the example of Panday and Huyakorn[2004], whoextended the 2-D V-catchment surface flow problem described in Section 4.2.3 to include thesubsurface domain, which extends to a depth of 20 m below the channel outlet. The domainis discretized into 11 subsurface layers with areal gridding mirroring the grid of the overlandflow domain. The top ten subsurface layers each have a thickness of 1 m and the bottomof the last layer is at z=-20 m. Again, only one half of the domain need be simulated dueto symmetry, with published outflow discharge halved, to produce equivalent results. Thesubsurface domain has horizontal and vertical hydraulic conductivity values of 5× 10−5 and5× 10−6 m s−1 respectively, a porosity value of 0.1, and Van Genuchten parameters α, βand Swr equal to 2.25 m−1, 1.89 and 0.16 respectively.

The same rainfall intensity of 3× 10−6 m s−1 as for the previous case is applied for a periodof 35 days, followed by no rainfall for the second stress period of 15 days.

In order to gauge the performance of the evapotranspiration formulation, evapotranspirationlosses, as parameterized in Table 4.11 were applied. Because Modflow-Surfact uses a block-centred approach while HydroGeoSphere uses a node-centred approach, the evaporationand root zone distribution functions are similar, but not identical.

Adaptive time stepping with an initial time-step size of 5s, a maximum time-step size of10000s and a time-step incrementing factor of 2.0 is used for the simulation. Newton iterationconsiderations include a maximum of 15 iterations and an absolute convergence tolerance of10−3.

Figure 4.33 compares predicted fluid fluxes from HydroGeoSphere with predictions fromModflow-Surfact. The responses of the two models are somewhat similar. The disparity isattributed to differences in the formulation of the surface/subsurface flux coupling termbetween the block-centred and node-centred approaches.

4.5 Subsurface Transport

4.5.1 Level 1: Chain Decay Transport in a Porous Medium

In order to verify the accuracy of HydroGeoSphere in simulating the movement of amulti-component decay chain in a porous medium it was compared with the exact analyticalsolution CMM [Sudicky, 1991; Sudicky et al., 2013].


Table 4.11: Parameter Values for the Simulation of the Panday and Huyakorn[2004] Study.Parameter Value UnitReference evapotranspiration 3× 10−7 mLeaf area index 2.08Saturation at field capacity 0.32Saturation at wilting point 0.2Saturation at oxic limit 0.76Saturation at anoxic limit 0.9Energy limiting stage saturation 0.32Evaporation limiting stage saturation 0.2Transpiration constants

C1 0.3C2 0.2C3 3× 10−6 m s−1

Soil evaporation distribution functionDepth = 0.0 0.703Depth = 1.0 0.259Depth = 2.0 0.037Depth = 3.0 0.0

Root zone distribution functionDepth = 0.0 0.703Depth = 1.0 0.259Depth = 2.0 0.037Depth = 3.0 0.0


Time(seconds)

Flu

idflu

x(m

3 /sec

ond)

0 1E+06 2E+06 3E+06 4E+0610-2

10-1

100

101DischargeInfiltrationExfiltrationEvapotranspiration

Modflow-Surfact

10-2

10-1

100

101DischargeInfiltrationExfiltrationEvapotranspiration

HydroGeoSphere

Figure 4.33: Water Budget Components for the Simulation of the Panday andHuyakorn.[2004] Study.


Table 4.12: Parameter Values for Chain-decay Transport in a Porous Medium.

Parameter Value UnitVelocity 100.0 m yr−1

Dispersivity 10.0 mDiffusion coefficient 0.0 m2 yr−1

Retardation factorUranium234 1.43× 104

Thorium230 5.0× 104

Radium226 5.0× 102

Decay coefficientUranium234 2.83× 10−6 yr−1

Thorium230 9.0× 10−6 yr−1

Radium226 4.33× 10−6 yr−1

Initial source concentrationUranium234 1.0Thorium230 0.0Radium226 0.0

The problem was set up for the three-member decay chain:

Uranium234 → Thorium230 → Radium226

The input parameters and values for the analytical solution are shown in Table 4.12:

The analytical solution can simulate 1-D, 2-D or 3-D behavior with respect to plumedevelopment. In this case, it was set up to simulate 1-D behavior.

In the numerical model, a grid which was 500 m long in the x-direction and 1 m in they and z directions was generated. A uniform nodal spacing of 5 m was used in the x-direction. Medium properties and flow boundary conditions were assigned such that auniform linear flow velocity equal to 100 m yr−1 parallel to the x-axis was produced. Aconstant concentration of 1.0 was specified for Uranium234 at the upstream x-face.

Figure 4.34 shows concentration profiles for the three members of the decay chain at a timeof 10000 years for both CMM and HydroGeoSphere. It can be seen that the results arealmost identical.

4.5.2 Level 1: Chain Decay Transport in a Single Fracture

In order to verify the accuracy of HydroGeoSphere in simulating the movement of amulti-component decay chain in a fractured media it was compared with the exact analyticalsolution DKCRACK [Sudicky, personal communication, 1994].

The problem involves the same three-member decay chain used in the porous media exampledescribed above. The input parameters and values for the analytical solution are shown inTable 4.13.


Table 4.13: Parameter Values for Chain-decay Transport in a Fracture.Parameter Value UnitVelocity in fracture 100.0 m yr−1

Longitudinal dispersivity in fracture 1.0 mFracture aperture 1.0× 10−4 mFracture separation 0.1 mMatrix porosity 0.01Matrix tortuosity 0.1Inlet velocity † 100.0 m yr−1

Inlet dispersion 0.0 m yr−1

Diffusion coefficient ‡Uranium234 3.1536× 10−2 m2 yr−1

Thorium230 3.1536× 10−2 m2 yr−1

Radium226 3.1536× 10−2 m2 yr−1

Fracture retardation factorUranium234 1.0Thorium230 1.0Radium226 1.0

Matrix retardation factorUranium234 1.43× 104

Thorium230 5.0× 104

Radium226 5.0× 102

Decay coefficientUranium234 2.83× 10−6 yr−1

Thorium230 9.0× 10−6 yr−1

Radium226 4.33× 10−6 yr−1

Initial source concentrationUranium234 1.0Thorium230 0.0Radium226 0.0

† An inlet velocity equal to the velocity in the fracture and an inlet dispersion of 0 isequivalent to a first-type source‡ Free solution diffusion coefficient


Distance (m)

C/C

o

0 100 200 300 400 50010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

UraniumThoriumRadium

CMM

0 100 200 300 400 50010-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


HydroGeoSphere

Figure 4.34: Results for a 3-member Decay Chain in a Porous Medium at 10000 Years.

Note that subsurface water velocity in the matrix is 0.0, an assumption made in the analyticalsolution. Movement of contaminant into the matrix blocks is solely by molecular diffusion.

In the numerical model, a grid which was 200 m long in the x-direction and 1 m in the yand 0.1 m in the z directions was generated. A uniform nodal spacing of 5 m was used inthe x-direction. Medium properties and flow boundary conditions were set up such that auniform fracture flow velocity of 100 m yr−1 parallel to the x-axis was produced. A constantconcentration of 1.0 was specified for Uranium234 at the upstream end of the fracture.

Figure 4.35 shows the concentration profiles of Uranium, Thorium and Radium at 10000years for both DKCRACK and HydroGeoSphere simulations. The results are nearlyidentical.

4.5.3 Level 1: Time-variable Source Condition

In order to verify the accuracy of HydroGeoSphere in simulating a time-variable sourcecondition, it was compared with an exact analytical solution SUPER1D [Sudicky, personalcommunication, 1986] for a uniform vertical, one-dimensional flow field. A time-variableTritium source is applied with an with an input function for Tritium Units (TU) as shownin Figure 4.36. Radioactive decay is neglected in the simulation. Other input parameters forthe analytical solution are shown in Table 4.14.


Distance (m)

C/C

o

0 50 100 150 20010-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


DKCRACK

0 50 100 150 20010-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


HydroGeoSphere

Figure 4.35: Results for a 3-member Decay Chain in a Fractured Medium at 10000 Years.

Time(years)

Con

cent

ratio

n(T

U)

0 10 20 30 40100

101

102

103

104

Figure 4.36: Input Function for Time-variable Source Transport.


Table 4.14: Parameter Values for Time-varying Source Transport Simulation.Parameter Value UnitVelocity 1.0 m yr−1

Dispersivity 1.0 mDiffusion coefficient 0.0 m2 yr−1

Retardation factor 1.0

Distance (m)

C/C

o

0 10 20 30 40 50 60

10-2

10-1

100

101

102

103

7 years24 years

SUPER1D

0 10 20 30 40 50 60

10-2

10-1

100

101

102

103

7.0000000024.0000000

HydroGeoSphere

Figure 4.37: Results for a Time-variable Source Function.

In the numerical model, a grid which was 500 m long in the x-direction and 1 m in the yand z directions was generated. A uniform nodal spacing of 5 m was used in the x-direction.Medium properties and flow boundary conditions were set up such that a uniform averagelinear subsurface water flow velocity of 1.0 m yr−1 parallel to the x-axis was produced. Thetime-variable source function given above was specified at the upstream x-face.

Concentration profiles for the contaminant at times equal to 7 and 24 years are shown inFigure 4.37. Results from HydroGeoSphere are nearly identical to the analytical solutions.

In the numerical model, the adaptive timestepping routine was used and the maximumpercent concentration change allowed was 10%. The results are very close to those obtainedfrom the analytical solution.


Table 4.15: Parameter Values for Dual-porosity Transport Simulation.Parameter Value UnitDarcy velocity 2.5 cm day−1

Total porosity 0.5Mobile fraction 0.5Dispersion coefficient 10.0 cm2 day−1

Retardation factor 1.0Mass transfer coefficient 0.1 day−1

4.5.4 Level 1: Transport in a Dual-Porosity Medium

In order to verify the accuracy of HydroGeoSphere in simulating transport in a dual-porosity medium, it was compared with the analytical solution MPNE [Chris Neville,personal communication]. The input parameters for the analytical solution are shown inTable 4.15.

In the numerical model, a grid which was 60 cm long in the x-direction and 1 cm in the yand z directions was generated. A uniform nodal spacing of 1 cm was used in the x-direction.Medium properties and flow boundary conditions were set up such that a uniform averagelinear flow velocity of 10 cm day−1 parallel to the x-axis resulted. A constant concentrationof 1.0 was specified at the upstream end of the system. The porosity of both the mobile andimmobile zones was set to 0.25, which gives a total porosity of 0.5 with the mobile fractionbeing equal to 0.5, which is equivalent to the analytical solution input parameters. Thedispersivity was set to 1.0 cm, which is equivalent to a dispersion coefficient of 10 cm2 day−1

when multiplied by the average linear subsurface water velocity.

Figure 4.38 shows a concentration profile at a time of 2.5 days for both MPNE andHydroGeoSphere. It can be seen that the results are almost identical.

4.5.5 Level 2: Coupled Flow and Transport in a Dual-Permeability Medium

The capability of HydroGeoSphere to simulate transient flow and solute transport ina coupled porous medium-dual continuum system was tested by comparing it to resultsobtained by Gerke and Van Genuchten [1993]. The problem consists of a one-dimensionalcolumn 40 cm in length, in which the macropores are assumed to be planes with a spacingof 2 cm that subdivide the porous medium into uniform blocks. Similar to Gerke and VanGenuchten [1993], the domain was finely discretized in the vertical direction, using elementswhich were 0.1 m in length. The initial pressure head in the domain was set to -1000 cm. Auniform rainfall of 50 cm day−1 was applied to the dual continuum, and water was allowedto drain freely from the base of the system in both the porous medium and dual-continuum.The initial solute concentration in the domain was set to 1.0. The hydraulic and transportparameters of the system are presented in Table 4.16. The fluid exchange term αwd is0.6 cm−1 day−1, which is half that used by Gerke and Van Genuchten [1993]. This is away of reconciling the results of the two codes, as Gerke and Van Genuchten [1993] use


Distance (m)

C/C

o

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

MPNE

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

HydroGeoSphere

Figure 4.38: Results for Transport in a Dual-porosity Medium.

central weighting of relative permeability for computing the exchange fluxes between thetwo continua, while HydroGeoSphere uses upstream weighting.

Figure 4.39 shows pressure head profiles versus depth at 0.01, 0.04 and 0.08 days for boththe porous medium and dual continuum.

Figure 4.40 shows concentration profiles versus depth at 0.01, 0.04 and 0.08 days for boththe porous medium and dual continuum.

4.5.6 Level 2: Transport Due to an Injection/Withdrawal Well

This verification problem considers an injection/pumping cycle for a fully penetrating wellin a confined aquifer (Figure 4.41). Water with a constant concentration Co is injectedinto the well in the center of the domain for the first stress period. For the second stressperiod, the flow is reversed and the contaminated water is pumped out. The system reachesa steady state instantaneously for each stress period. An approximate analytical solution forthis problem is given by Gelhar and Collins [1971].

The numerical model consist of 31 columns, 31 rows and one layer. All parameters arepresented in Table 4.17. A constant head of 50 feet was applied on the outer boundary of thedomain to produce a steady state flow-field. The simulation was done in two periods: thefirst stress period is an injection for 2.5 years and the second stress period is a pumping for


Table 4.16: Parameters for the Gerke and Van Genuchten [1993] study.Parameter Value UnitPorous medium

Saturated hydraulic conductivity, K 1.0526 cm day−1

Porosity, θs 0.5Specific storage, Ss 1× 10−7 cm−1

Volume fraction, wm 0.95Van Genuchten α 0.005 cm−1

Van Genuchten β 1.5Residual water saturation Swr 0.10526Dispersivity αl, αt 2.0 cm

Dual continuumSaturated hydraulic conductivity, Kd 2000 cm day−1

Porosity, θsd 0.5Specific storage, Ssd 1× 10−7 cm−1

Volume fraction, wd 0.05Van Genuchten α 0.01 cm−1

Van Genuchten β 2.0Residual water saturation Swrd 0.0Dispersivity αld, αtd 2.0 cm

InterfaceSaturated hydraulic conductivity, Ka 0.01 cm day−1

Geometrical shape factor βd 3Van Genuchten α 0.005 cm−1

Van Genuchten β 1.5Residual water saturation Swr 0.10526Fluid exchange term αwd 0.6 cm−1 day−1

Mass exchange term αs 0.15 day−1


Pressure head (cm)

Dep

th(c

m)

-1000 -800 -600 -400 -200 0

0

10

20

30

40

Matrix 0.01 daysMatrix 0.04 daysMatrix 0.08 daysFracture 0.01 daysFracture 0.04 daysFracture 0.08 days

Gerke and van Genuchten (1993)


HydroGeoSphere

Figure 4.39: Pressure head profiles of the Gerke and Van Genuchten [1993] study.

Solute Concentration

Dep

th(c

m)

0 0.2 0.4 0.6 0.8 1

0

10

20

30

40


Gerke and van Genuchten (1993)


HydroGeoSphere

Figure 4.40: Concentration profiles of the Gerke and Van Genuchten [1993] study.


t1t2 t3

t4

Withdrawal period2.5 years < t <t < 10 years3 4

Injection period0< t <t < 2.5 years1 2

Figure 4.41: Injection/withdrawal Well System.

a period of 7.5 years. The first stress period was divided into 10 time steps and the second,into 28 time steps. The simulation with HydroGeoSphere is compared with the analyticalsolution in Figure 4.42. The breakthrough curve obtained from HydroGeoSphere iscomparable to the analytical solution.

4.5.7 Level 1: Two-Dimensional Transport from a Point Source in aSteady State Uniform Flow Field

This problem concerns two-dimensional dispersion of solute in a uniform and steady subsur-face water flow field as depicted in Figure 4.43. It assumes a small injected rate to avoiddisturbance of the natural subsurface water flow field. An analytical solution for such asituation can be found in Wilson and Miller [1978].

Two cases involving conservative and non-conservative species were simulated using Hydro-GeoSphere. The model parameters, shown in Table 4.18, are taken from Huyakorn et al.[1984b]. The selected parameter values for Case 1 were based on data from the field study ofhexavalent chromium contamination reported by Perlmutter and Lieber [1970]. For Case 2,the values of retardation and decay constant were chosen arbitrarily to test the performanceof HydroGeoSphere for a non-conservative species.

Both simulations were performed using a rectangular domain containing 741 30 m × 30 melements and 14 equal time steps of 100 days each. Concentration profiles along the x-axisat t=1400 days are plotted in Figure 4.44. Numerical results from HydroGeoSphere arecompared with the analytical solution, the finite-difference model MODFLOW-SURFACT


Table 4.17: Model Parameters for the Simulation of Transport From an Injection/extractionWell.

Parameter Value UnitCell width along rows ∆x 900 ftCell width along columns ∆y 900 ftThickness 20 ftHydraulic conductivity of the aquifer 0.005 ft s−1

Porosity 0.3Longitudinal dispersivity αL 100 ftTransverse dispersivity αT 100 ftVolumetric injection (+)/extraction(-) rate for first/second stress periods 1 ft3 s−1

Concentration during the first stress period 1Length of the injection period 2.5 yrLength of the extraction period 7.5 yr

Time since pumping (years)

Rel

ativ

eco

ncen

trat

ion

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Gelhar & Collins (1971)

0 1 2 3 4 5 6 7 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

HydroGeoSphere

Figure 4.42: Breakthrough Curve from Simulation of Transport Due to an Injec-tion/withdrawal Well.


570 m

1170 m

Ambient groundwater flow

Source

Figure 4.43: Two-dimensional Transport from a Point Source.

Table 4.18: Parameters for Simulation of 2-D Transport from a Point SourceParameter Value UnitDarcy velocity q 0.161 m day−1

Porosity θ 0.35Longitudinal dispersivity αL 21.3 mTransverse dispersivity αT 4.3 mThickness of the saturated aquifer b 33.5 mContaminant Mass flux per unit thickness of aquifer Qco 704 g m−1 day−1

CASE 1:Linear adsorption coefficient, Kd 0 m3 kg−1

Retardation coefficient, R = ρbKd/ϕ 1Decay constant, λ 0 day−1

CASE 2:Linear adsorption coefficient, Kd 0.14 m3 kg−1

Retardation coefficient R = ρbKd/ϕ 2Decay constant λ 0.00019 day−1

Bulk density ρb 2.5 kg m−3


Distance from source (m)

Con

cent

ratio

n(g

/m^3

)

0 200 400 600 800 10000

20

40

60

80

100Wilson & Miller (1978)

MODFLOW-SURFACT (TVD)

FTWORK (central)HydroGeoSphere

Case 1

Case 2

Figure 4.44: Concentration Profiles Along the Center Line for t =1400 Days.

[HydroGeoLogic, 1996], and the finite-difference solution of FTWORK [Faust et al., 1993].In each case HydroGeoSphere produces solutions which are of comparable accuracy withrespect to the other models.

4.5.8 Level 1: Transport Due to an Injection-Withdrawal Well Pair

This problem concerns solute transport between a pair of discharging and recharging wellsoperating at a constant flow rate. Both wells fully penetrate a constant-thickness confinedaquifer that is assumed to be homogeneous, isotropic and of infinite areal extent. The flowfield is assumed to be in steady-state with typical flow lines and solute fronts depicted inFigure 4.45.

The analytical solution which represents this case was developed by Hoopes and Harleman[1967]. The numerical model consist of 31 columns, 33 rows and one layer. Model parametersare presented in Table 4.19. The hydraulic conductivity used is 0.005 cm s−1.

The simulation was done with fifty time steps for a total simulation of 8 × 104 s. Theinitial time step size is 2 s and increases by a factor of 1.2 thorough the simulation. Theinitial concentration in the domain is equal to zero and a unit concentration is prescribedat the injection well. Figure 4.46 presents the results obtained from HydroGeoSpherewhich compares well with the analytical solution. Breakthrough is however diffused in thenumerical solution resulting from the coarse discretization used in a complex flow field.


Figure 4.45: Injection-withdrawal Well Pair.

Table 4.19: Parameters for Simulations of Transport Due to an Injection-withdrawal Pair.Parameter Value UnitWell flow rate, Qinj=Qpump 2.339 cm3 s−1

Well spacing 61.0 cmThickness of aquifer, b 8.9 cmPorosity, 0.374Retardation coefficient, R 1Decay constant, λ 0Longitudinal dispersivity αL 0.294 cmTransverse dispersivity αT 0


Time (seconds)

Rel

ativ

eC

once

ntra

tion

103 104 1050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Hoopes and Harlemann [1967]

103 104 1050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

HydroGeoSphere

Figure 4.46: Breakthrough Curve of Concentration Solute at the Pumping Well.

4.5.9 Level 2: Two-Dimensional (Areal) Transport of a ContaminantPlume in a Heterogeneous Confined Aquifer with a Pair of InjectionAnd Withdrawal Wells And Strong Ambient Subsurface Flow

This problem taken from Zheng [1990] is shown in Figure 4.47. It concerns contaminanttransport in a heterogeneous aquifer under a strong ambient steady-state subsurface waterflow field. North and South boundaries represent zero flux conditions while East and Westboundaries represent constant-head conditions whereby the East boundary has also animposed gradient. Contaminant is injected into the aquifer for a period of one year and issubsequently allowed to distribute for another five years. The total mass injected is 1825 kg(Mi=Qi Ci ti). One well located downstream pumps 0.0189 m3/s throughout the simulationperiod. A low conductivity zone is located between the two wells (200 m × 600 m). Forboth zones, longitudinal and transverse dispersivities are respectively equal to 20 et 4 m.The effective porosity is equal to 0.3.

The grid used for HydroGeoSphere simulations contains 18 columns, 14 rows and onelayer of cells of uniform dimensions of 100 m × 100 m. The simulation was done with 100equal time steps for a total time of 6 years. Results show a good correlation between theHydroGeoSphere simulation and MT3D results (see Figure 4.48) with negligible massbalance error as shown in Figure 4.49.


Injection well

Q = 0.001 m /s

C =57.87 ppm

= 1 year

i

i

i

3

t

Low

con

du

ctiv

ity

zon

e

K=

1.4

74 E

-7 m

/s

Pumping well

Q = 0.0189 m /s

6 years

p

3

t =p

Confined aquifer: thickness =10 m

K= 1.474 E-4 m/s

Con

stan

t H

ead

: h

=250 m

h=

18.7

5 m

Gra

die

nt

2.5

/100

H=

53.7

5 m

No mass flux

No mass flux

No m

ass

flu

x

1400 m

1800 m

N 200 m

600 m

Figure 4.47: Problem Description for 2-D Transport in a Heterogeneous Confined Aquifer.


Time (years)

Con

cent

ratio

n(k

g/m

^3)

0 1 2 3 4 5

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001 MT3D Zheng [1990]

0 1 2 3 4 5

-0.0002

0

0.0002

0.0004

0.0006

0.0008

0.001HydroGeoSphere

Figure 4.48: Concentrations Observed at the Pumping Well.

Time

Mas

s(K

g)

1 2 3 4 5 6-2000

-1500

-1000

-500

0

500

1000

1500

2000

In (cumulative)Out (cumulative)StoredIn+Out-Stored

Figure 4.49: Mass Budget for the 2-D Transport Simulation.


Injection well

Q = 0.15 m /s

C =100 ppb

= 1 year

i

i

i

3

t

Low

con

du

ctiv

ity

zon

e

K=

1.4

74 E

-6 m

/s

Pumping well

Q = Transientp

Confined aquifer: Total thickness = 25 m

K= 1.474 E-3m/s

S=0.01

Con

stan

t h

ead

: h

=20 m

No mass flux

No mass flux

No m

ass

flu

x

Con

stan

t h

ead

: h

=16 m

1400 m

1800 m

Figure 4.50: Problem Description for 2-D Transport.

4.5.10 Level 2: Two-Dimensional Transport of a Contaminant Plume ina Heterogeneous Confined Aquifer

This problem has the same setting as the one presented in Section 4.5.9 except that thesimulation is done within a transient flow field. The total thickness of the aquifer is 25m. The set-up is shown in Figure 4.50. The total mass injected is 473.05 kg. The specificstorage is estimated to be 0.01. For both zones, the porosity, the longitudinal and transversedispersivities are equal respectively to 0.25, 20 m and 4 m.

The simulation was done in transient flow for a period of 7 years. The grid used is the sameas the one in the example shown in Section 4.5.9. The pumping rate is 0.24 m3 s−1 for a 1year period separated by 1 year without pumping. Pumping begins after one year and thetotal duration of simulation is seven years which includes 3 pumping stages. The initial timestep size is 1.2 days which is incremented by a factor of 1.5 up to a maximum time step sizeof 12 days. Hydraulic heads observed at the pumping well are presented in Figure 4.51. Thecomparison is made with MODFLOW.

Solute concentrations at the pumping well were evaluated during the simulation and arepresented in Figure 4.52. HydroGeoSphere results compare fairly well with the break-


Time (years)

Hyd

rau

licH

ead

(m)

1 2 3 4 5 6 712

13

14

15

16

17

18

19

MT3D Zheng [1990]

1 2 3 4 5 6 712

13

14

15

16

17

18

19

HydroGeoSphere

Figure 4.51: Hydraulic Head Distribution at the Pumping Well During the Simulation.

through obtained from MT3D. The solute mass balance evaluated in Figure 4.53 showsnegligible mass balance errors throughout the simulation. The total mass OUT is about 474g, a little more than the total mass IN which is 473.05g.

4.5.11 Level 2: Two-Dimensional Transport of Contaminant in the WaterPhase of an Unsaturated Rectangular Soil Slab

This problem concerns transport of a non-conservative solute in a transient 2-D unsaturatedflow field. The system is initially dry and free of solutes and water with dissolved soluteis allowed to enter the system at the upper portion of the left hand boundary as shown inFigure 4.54. Inflow head is 6 cm with a prescribed solute concentration of 1 ppm. Outflowoccurs along the entire right hand side boundary under initial pressure conditions of -90 cm.The remaining boundaries are under no-flow, zero concentration gradient conditions.

The domain dimension is 15 cm horizontally and 10 cm vertically, discretized using a gridwith constant cell spacing of 1cm. The soil is assumed to be homogeneous and isotropic.Table 4.20 presents the hydraulic properties of the soil. The simulation was done with aninitial time step size of 0.01 days which is enlarged with a factor of 1.2 up to a maximumtime step size of 0.05 days. The physical transport parameters are shown in Table 4.21. Thetotal duration of the simulation is 0.508 days. The values of concentration in the rectangularsoil slab are computed for t=0.508 days and are presented in Figure 4.55.


Time (years)

Con

cent

ratio

n(k

g/m

3)

0 1 2 3 4 5 60

2E-05

4E-05

6E-05

8E-05

0.0001

0.00012

MT3D Zheng [1990]

0 1 2 3 4 5 60

2E-05

4E-05

6E-05

8E-05

0.0001

0.00012

HydroGeoSphere

Figure 4.52: Breakthrough Curve Observed at the Pumping Well for 2-D Transport Simula-tion.

Time (years)

Mas

s(K

g)

1 2 3 4 5 6

-400

-200

0

200

400 In (cumulative)Out (cumulative)StoredIn+Out-Stored

Figure 4.53: Solute Mass Balance for 2-D Transport Simulation in Transient Flow Field.


4 c

m6

cm

15 cm

No mass flux

No m

ass

flu

x

No m

ass

flu

xC=1 for 6< y < 10

y

x

No mass flux

Figure 4.54: Problem Description for 2-D Transport in an Unsaturated Rectangular SoilSlab.

Table 4.20: Hydraulic Properties of the Rectangular Soil Slab.Parameter Value UnitSaturated hydraulic conductivity, K 1 cm day−1

Specific storage, Ss 1× 10−14

Van Genuchten α 0.005 cm−1

Van Genuchten β 2Brooks Corey parameter, n 2Residual water saturation Swr 0.3333

Table 4.21: Physical Parameters Values for Simulation of Transport in an UnsaturatedRectangular Soil Slab.

Parameter Value UnitPorosity 0.45Initial concentration Co 0Longitudinal dispersivity αL 1 cmTransverse dispersivity αT 0Molecular diffusion Do 0.01 cm2 day−1

Decay constant 0.001 day−1

Bulk density 1.46 g cm−3

Distribution coefficient Kd 0.308 cm3 g−1


Distance (m)

Ele

vatio

n(m

)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

4

5

6

7

8

9

10

MODFLOW-SURFACTSTAFF3DHYDROSPHERE

0.9

0.8

0.70.6

0.50.4

0.3

0.2

0.1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

1

2

3

4

5

6

7

8

9

10

Figure 4.55: Simulated Contaminant Concentrations in an Unsaturated Rectangular SoilSlab at 0.508d.


Table 4.22: Parameters used for the Saltpool 1 SimulationParameter Value UnitFree-solution diffusion coefficient (Dd) 3.56× 10−6 m2 s−1

Brine density (ρmax) 1200 kg m−3

Reference density (ρ0) 1000 kg m−3

Fluid dynamic viscosity (µ) 1.124× 10−3 kg m−1 s−1

Hydraulic conductivity (K) 150 m yr−1Porosity (n) 0.1Longitudinal dispersivity (αL) 0.0 mTransverse dispersivity (αT ) 0.0 m

4.6 Variable-Density Flow

4.6.1 Level 2: Variable-Density Flow in Porous Media, Elder’s Problem

Variable-density flow and transport in porous media was verified in two dimensions using theElder [1967] salt convection problem. The results presented by Frolkovic and De Schepper[2000] were chosen to be compared to those of HydroGeoSphere, and are considered moretrustworthy since both numerical models use the control volume finite element method aswell as the same flow variable (fluid pressure, P ).

Frolkovic and De Schepper [2000] carried out their numerical simulations in the half domainof the symmetric Elder problem, using an extremely fine grid consisting of 32,768 nodes. Allsimulations used implicit transport time weighting, as is common in other variable-densitysimulations, and full upstream weighting as proposed by Frolkovic and De Schepper [2000].

The physical parameters given by Oswald and Kinzelbach [2004] were used and are shown inTable 4.22.

Their results are in very good visual agreement with those from the HydroGeoSpheremodel (Figure 4.56).

4.6.2 Level 3: Variable-Density Flow in Porous Media, Saltpool Experi-ment

A new benchmark problem for variable-density transport in 3D has been presented by Oswaldand Kinzelbach [2004]. This problem is based on the three-dimensional variable-density flowand solute transport experiments in porous media conducted by Oswald [1999]. In theseexperiments, a 0.2 m × 0.2 m × 0.2 m closed box initially contained saltwater from thebottom up to 6 cm, with the rest of the box filled with freshwater. A constant freshwaterrecharge through one upper corner of the box disturbed this stable layering of two misciblefluids. The concentration of the mixed fluid versus time was measured at the dischargingopen hole on the opposite side of the input location. Oswald [1999] used two different initialconcentrations c01 = 0.01 (case 1) and c02 = 0.1 (case 2). The experimental results were


Horizontal Distance (m)

Ele

vatio

n(m

)

0 50 100 150 200 250 3000

50

100

150

Frolkovic et. al. [2000]HydroGeoSphere

0 50 100 150 200 250 3000

50

100

150


Ele

vatio

n(m

)

0 50 100 150 200 250 3000

50

100

150


0 50 100 150 200 250 3000

50

100

150


Ele

vatio

n(m

)

0 50 100 150 200 250 3000

50

100

150


0 50 100 150 200 250 3000

50

100

150


Ele

vatio

n(m

)

0 50 100 150 200 250 3000

50

100

150


0 50 100 150 200 250 3000

50

100

150

Figure 4.56: Results of the Elder problem for an extremely fine grid (256 × 128 elements inthe half domain) at 2.5, 5, 10 and 20 years simulation time. Shown are the 20%, 40%, 60%and 80% contours.


Table 4.23: Parameters used for the Saltpool 1 SimulationParameter Value UnitFree-solution diffusion coefficient (Dd) 8.7× 10−10 m2 s−1

Brine density (ρmax) 1005.9 kg m−3

Reference density (ρ0) 998.23 kg m−3

Fluid dynamic viscosity (µ) 1.002× 10−3 kg m−1 s−1

Hydraulic conductivity (K) 9.325× 10−3 m s−1Porosity (n) 0.372Longitudinal dispersivity (αL) 0.0012 mTransverse dispersivity (αT ) 0.00012 m

numerically reproduced by Johannsen et al. [2002], who also present tabular data of themeasured concentrations versus time.

The HydroGeoSphere model output was compared in three dimensions with experimentalresults of Oswald [1999], given in Johannsen et al. [2002]. The physical parameters given byOswald and Kinzelbach [2004] were used and are shown in Table 4.23.

The first problem of the lower initial concentration 0.01 (case 1) was used because Johannsenet al. [2002] showed that, in this case, grid convergence is achieved with a relatively coarsegrid, whereas for case 2, the solution converged only for a very fine grid, consisting of at least274,625 grid points [Johannsen et al., 2002]). Good agreement between the experimentalresults from Oswald [1999], the numerical results from Diersch and Kolditz [2002] and theHydroGeoSphere model was obtained (Figure 4.57). The long-term results of this lowdensity case more closely resemble the experimental data than in Diersch and Kolditz [2002];however, differences remain.

4.6.3 Level 2: Variable-Density Flow in Fractured Porous Media

Variable-density flow in vertical fractures was verified by reproducing the results presentedby Shikaze et al. [1998]. The trial which includes only vertical fractures was used as a testcase. The external hydraulic heads on both aquifer top and bottom were set to zero becauseShikaze et al. [1998] showed that density effects are best accounted for if the imposed headgradient vanishes. Otherwise, the effect of forced convection may suppress free convection. Inthe numerical simulations, the left and right boundaries were assumed to be impermeable forflow. The top of the domain is assumed to be a salt lake with a constant concentration equalto 1.0. All other boundaries for transport are zero dispersive-flux boundaries. The physicalparameters used are identical to those presented in Shikaze et al. [1998] and summarizedin Table 4.24. The 3D domain is of size `x = 10 m, `y = 1 m and `z = 10 m. Thespatial discretization used was 0.025 m in both the x- and the z-direction and unity in they-direction. Fracture spacings are nonuniform as shown in Figure 4.58. The Figure showsexcellent agreement between the concentration distributions calculated by the two numericalmodels.


Time (seconds)

Sal

tmas

sfr

actio

n(%

)

0 2000 4000 6000 80000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

Experimental (Johannsen 2002)Numerical (Diersch and Kolditz, 2002)

0 2000 4000 6000 80000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

HydroGeoSphere

Figure 4.57: Results of three-dimensional variable-density transport simulations in porousmedia.

Table 4.24: Model parameters used in fractured media studies. All parameters are identicalto those used by Shikaze et al. [1998].

Parameter Value UnitFree-solution diffusion coefficient (Dd) 5×10−9 m2 s−1

Brine density (ρmax) 1200 kg m−3

Reference density (ρ0) 1000 kg m−3

Fluid compressibility (αfl) 4.4×10−10 kg−1 m s2

Matrix compressibility (αm) 1.0×10−8 kg−1 m s2

Fluid dynamic viscosity (µ) 1.1×10−3 kg m−1 s−1

Matrix permeability (κij) 10−15 m2

Matrix longitudinal dispersivity (αl) 0.1 mMatrix transverse dispersivity (αt) 0.005 mMatrix porosity (φ) 0.35Tortuosity (τ) 0.1Fracture dispersivity (αfr) 0.1 mFracture aperture (2b) 50 µm



Ele

vatio

n(m

)

0 2 4 6 8 100

2

4

6

8

10

Shikaze et. al. [1997]

HydroGeoSphere

0 2 4 6 8 100

2

4

6

8

10

Figure 4.58: Variable-density flow in a set vertical fractures embedded in a porous matrix.Shown are the concentration contours 0.1 to 0.9 with a contour interval of 0.4 at 2 yearssimulation time.


4.7 Heat Transfer

4.7.1 Level 1, 2: Heat Transfer in Porous Media

The first test case verifies 1D heat transfer in an unfractured porous matrix. A constantvelocity along the flow axis is imposed. The impact of temperature on fluid properties isignored, which linearizes the problem. Thermal energy is transported by way of conduction,advection and mechanical dispersion. In this case, the governing equation can be written inthe form:

Dth∂2T

∂x2 − vth∂T

∂x= ∂T

∂t(4.6)

where Dth [L2 T−1] is the thermal dispersion coefficient:

Dth = kb + φDxxρlclρbcb

(4.7)

and vth [L T−1] is the retarded velocity:

vth = q · ρlclρbcb

= q · 1φRth

(4.8)

with the thermal retardation coefficient Rth [–] [Molson et al., 1992]:

Rth = 1 + (1− φ)ρscsφρlcl

(4.9)

Equation (4.6) has the standard parabolic-hyperbolic form of a 1D partial differentialequation. Therefore, if (4.6) is subject to the Dirichlet boundary condition, T = T1, thesolution is the Ogata-Banks analytical solution:

T − T0T1 − T0

= 12

[erfc

(x− vtht2√Dtht

)+ exp

(vthx

Dth

)erfc

(x+ vtht

2√Dtht

)](4.10)

where T0 is the initial temperature in the domain.

In the numerical simulation, the finite element domain was spatially discretized by using 20uniform blocks in the flow direction. All simulation parameters are given by Table 4.25. Thedeveloped numerical model is compared with the analytical solution (4.10) as well as withnumerical results presented by Ward et al. [1984] who used the code SWIFT. The resultsare depicted in Figure 4.59.

4.7.2 Level 1: Heat Transfer in Fractured Media

The second test case verifies advective-conductive-dispersive 1D heat transfer in a singlefracture embedded in an impermeable matrix. As in the previous case, a constant flowvelocity along the axis is imposed and fluid properties are kept constant in order to linearizethe problem. Hence, the governing equation simplifies to:

Dfrth

∂2T fr

∂z2 − vfr ∂Tfr

∂z= ∂T fr

∂t(4.11)


0 100 200 300 400 500 6000

20

40

60

80

100

Analytical

Distance (m)

Tem

tera

ture

(deg

ress

C)

0 100 200 300 400 500 6000

20

40

60

80

100

SWIFT (Ward et. al., 1984)

0 100 200 300 400 500 6000

20

40

60

80

100

HydroGeoSphere

Figure 4.59: Temperature profiles of 1D heat transfer in an unfractured porous matrix(example 1). Shown are the temperatures in the matrix at 2,148 (left) and 4,262 (right)days.


Table 4.25: Model parameters used in the verification example for 1D heat transfer in anunfractured porous matrix. All parameters are identical to those used by Ward et al. [1984].

Parameter Value UnitBulk thermal conductivity (kb) 2.16 kg m s−3 K−1

Heat capacity of solid (cs) 1254.682 m2 s−2 K−1

Solid density (ρs) 1602 kg m−3

Heat capacity of water (cl) 4185 m2 s−2 K−1

Fluid density (ρl) 1000 kg m−3

Matrix porosity (φ) 0.1Longitudinal dispersivity (αl) 14.4 mHeat dispersion coefficienta (Dth) 1.15×10−5 m2 s−1

Thermal retardation coefficientb (Rth) 5.323Darcy flux (q) 3.53×10−7 m s−1

Retarded velocityc (vth) 6.63×10−7 m s−1

Initial temperature (T0) 37.78 oCBoundary temperature (T1) 93.33 oCDomain size (`x) 600 mOutput times (t1, t2) 2148, 4262 daysa from relation (4.7)b from relation (4.9)c q/(φRth)

where Dfrth [L2 T−1] is the fracture thermal dispersion coefficient, given by:

Dfrth = kl

ρlcl+Dfr

zz (4.12)

and where vfr is the constant groundwater flow velocity along the fracture. Equation (4.11)is a standard 1D parabolic-hyperbolic partial differential equation. If the Dirichlet boundarycondition T fr = T fr1 is imposed on the fracture inlet, the Ogata-Banks analytical solution isnow:

T fr − T fr0

T fr1 − Tfr0

= 12

erfc

z − vfrt2√Dfrth t

+ exp(vfrz

Dfrth

)erfc

z + vfrt

2√Dfrth t

(4.13)

where T fr0 is the initial temperature in the fracture.

In the numerical simulation, the finite element domain was spatially discretized along thefracture by using element sizes that gradually increase by the factor 1.1 from ∆z = 0.1 mnear the elevated temperature to ∆z = 15 m at the domain boundary. The groundwatervelocity in the fracture was set to 7.0×10−7 m s−1 and the fracture dispersivity used was5.0 m, giving the fracture thermal dispersion coefficient as 3.62×10−6 m2 s−1. All otherparameters are identical to those used in the previous example and given in Table 4.25. Thesimulation results are depicted in Figure 4.60.


Distance (m)

Tem

tera

ture

(deg

ress

C)

0 100 200 300 400 500 6000

20

40

60

80

100

Analytical

0 100 200 300 400 500 6000

20

40

60

80

100

HydroGeoSphere

Figure 4.60: Temperature profiles of 1D heat transfer in a single fracture within an imper-meable matrix (example 2). Shown are the temperatures in the fracture at 2,148 (left) and4,262 (right) days.


4.7.3 Level 1: Heat Transfer in Fractured Porous Media

The third test case verifies 2D heat transfer in a single fracture embedded in a porous matrix.This verification example is based on analytical results presented by Meyer [2004], whoinvestigated advective transient heat transfer in a fracture while in the porous matrix, heatis transported due to conduction alone. Mechanical heat dispersion as well as conductionwithin the fracture are not considered, making numerical integration unnecessary. Thegroundwater flow velocity in the fracture is constant. Under these assumptions, the governingequations of this problem simplify to:

ρbcb∂T

∂t− kb

∂2T

∂x2 = 0 b ≤ x ≤ ∞ (4.14)

and

ρlcl∂T fr

∂t+ ρlcl v

fr ∂Tfr

∂z− kb

b

∂T fr

∂x

∣∣∣∣∣x=b

= 0 0 ≤ z ≤ ∞ (4.15)

for heat transport in the matrix and in the discrete fracture, respectively. The last term in(4.15) expresses conductive loss of heat from the fracture into the matrix on the fracture-matrix interface. Initially, the entire system has the uniform temperature, T0. The fluidentering the fracture has the constant temperature, T1. All boundaries, except the fractureinlet and outlet, are impermeable for groundwater flow and for heat exchange. According toMeyer [2004], the transient solution along the fracture is:

T fr − T0T1 − T0

= erfc

z√kb ρbcb

2vfrρlclb√

(t− z/vfr)

(4.16)

Using the analytical results presented by Tang et al. [1981], it can be shown that the transientsolution along a cross-section from the fracture into the porous matrix is given by

T − T0T1 − T0

= erfc

z√kb ρbcb

2vfrρlclb√

(t− z/vfr)+

√ρbcb (x− b)

2√kb

√(t− z/vfr)

(4.17)

The fracture-matrix system used is identical to that shown in Figure 4.61. The finite elementdomain was spatially discretized in the x-direction by gradually increasing ∆x with constantfactor 1.1 from ∆x = 0.01 m near the fracture to ∆x = 0.1 m at the domain boundary. In theflow direction, ∆z also increases gradually from ∆z = 0.1 m near the elevated temperatureto ∆z = 0.5 m at the domain boundary. All other parameters are presented in Table 4.26and the simulation results are exhibited in the Figures 4.62 and 4.63.

4.7.4 Level 2: Heat Transfer in Anisotropic Porous Media

The last verification problem for heat transfer is the 2D field scale example presented byYang and Edwards [2000], and includes all heat transfer mechanisms in both continua with


vfr

(2 )b

x

z

FR

AC

TU

RE

PO

RO

US

MA

TR

IX

Figure 4.61: Fracture-matrix system used for model verification (Tang et al., 1981).


Distance (m)

Tem

per

atur

ein

Fra

ctur

e

0 2 410

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

Meyer (2004)

0 2 410

10.5

11

11.5

12

12.5

13

13.5

14

14.5

15

HydroGeoSphere

Figure 4.62: Temperature profiles of 1D heat transfer in discretely-fractured porous media.Shown are the temperatures in the fracture at 5,000 (left) and 10,000 (right) seconds.


Distance (m)

Tem

pera

ture

1 1.1 1.2 1.3 1.4 1.510

11

12

13

14

15

Meyer (2004)

1 1.1 1.2 1.3 1.4 1.510

11

12

13

14

15

1 1.1 1.2 1.3 1.4 1.510

11

12

13

14

15

HydroGeoSphere

Figure 4.63: Temperature profiles of 1D heat transfer in discretely-fractured porous media.Shown are the temperatures in the matrix at 10,000 seconds simulation time at the distances0.1 (left) and 0.61 (right) m from the fracture.


Table 4.26: Model parameters used in the verification example for 2D heat transfer in asingle fracture embedded in a porous matrix. All parameters are identical to those used byMeyer [2004].


Heat capacity of solid (cs) 908 m2 s−2 K−1



Fluid density (ρl) 997 kg m−3

Matrix porosity (φ) 0.2Groundwater flow velocity in the fracture (vfr) 0.05 m s−1

Initial temperature (T0) 10 oCBoundary temperature (T1) 15 oCDomain size (`x, `z) 2, 10 mLocation of cross-sections (z1, z2) 0.1, 0.61 mOutput times (t1, t2) 5000, 10000 s

variable fluid properties. This test case represents a realistic scenario of radioactive wastedisposal in the low-permeability anisotropic granitic rock of the Canadian Shield [Davison etal., 1994a]. Figure 4.64 shows the conceptual model, a vertical slice of dimensions 2,000 m× 1,000 m with a unit thickness. Radionuclides are disposed of in a 1,300 m long horizontalvault at a depth of 500 m below surface. The simulation domain consists of three anisotropicporous layers. The radioactive waste represents an exponentially decreasing heat source dueto remaining radioactivity [Davison et al., 1994a]. Thus, the term Γ = 11.59 kg m−1 s−3 ·exp(-5.5×10−10 s−1 · t), as given by Yang and Edwards [2000], was added as a heat sink termto the left hand side of the governing equation. All boundaries are impermeable for flow.Top and bottom boundaries have constant temperatures to mimic a geothermal gradient of11.5 K km−1, which is natural in the study area. All other boundaries are impermeable forheat transfer. Initially, the geothermal field is undisturbed with horizontal isotherms.

In the numerical simulations carried out with HydroGeoSphere, the temperature isassumed to have an impact on both fluid properties density and viscosity. This conformswith the assumption made by Yang and Edwards [2000]. Chemical reactions are notconsidered. All model parameters are summarized in Table 4.27. The variable-density,variable-viscosity flow and heat transfer results are exhibited in Figure 4.65, which showsexcellent agreement between the two numerical models.

4.7.5 Level 2: Borden thermal injection experiment

Subsurface thermal energy transport is verified by comparing results from HydroGeoSpherewith the model verification example used by Molson et al. [1992]. This verification exampleinvolves a simulation of the Borden thermal injection experiment. A description of theexperiment and the observed data are presented by Palmer et al. [1992]. Unless otherwise


Vault (heat source)

Layer 1

0

1000

500

500 1000 200015000

Layer 2

Layer 3

Figure 4.64: The conceptual model for variable-density heat transfer in anisotropic porousmedia (example 4; Yang and Edwards, 2000). The heat source in the vault is due to theremaining radioactivity of the stored waste. Top and bottom boundaries are assigned theconstant temperatures 6oC and 17.5oC, respectively, with the corresponding geothermalgradient 11.5 K km−1.

Table 4.27: Model parameters used in the verification example for 2D variable-densitythermal flow and heat transfer in anisotropic porous media. All parameters are identical tothose used by Yang and Edwards [2000].


Heat capacity of solid (cs) 800 m2 s−2 K−1



Matrix permeability (κxx, κzz) Layer 1: 1.0×10−15, 5.0×10−15 m2

Layer 2: 1.0×10−17, 5.0×10−17 m2

Layer 3: 1.0×10−19, 1.0×10−19 m2

Matrix porosity (φ) 0.004Domain size (`x, `z) 2000, 1000 mSpatial discretization (∆x,∆z) 25, 25 m



Ele

vatio

n(m

)

0 500 1000 1500 20000

200

400

600

800

1000

Yang et. al. [2000]HydroGeoSphere

20

200

0 500 1000 1500 20000

200

400

600

800

1000

(a)


Ele

vatio

n(m

)

0 500 1000 1500 20000

200

400

600

800

1000 8

26

44

62

80

0 500 1000 1500 20000

200

400

600

800

1000

(b)


Ele

vatio

n(m

)

0 500 1000 1500 20000

200

400

600

800

1000

8

10

12

14

16

0 500 1000 1500 20000

200

400

600

800

1000

(c)

Figure 4.65: Evolution of temperature in anisotropic porous media with an exponentiallydecreasing heat source. Simulation times are (a) 104 days, (b) 3×105 days, (c) 7×106 days.Shown are isotherms in degrees Celsius.


stated, all parameters used in the HydroGeoSphere simulation are identical to those usedby Molson et al. [1992]. The domain size was 40 × 30 × 20 m, and was discretized in alldirections using 0.5 m block elements. The initial temperature of the domain varied from15 oC at ground surface to 9 oC 6 m below the surface. The injection well was located atx = 12 m, y = 15 m, z = 16 m and water was injected at a temperature of 37 oC for thefirst 6 days. The flow and transport parameters are given in Table I. The aquifer thermalparameters are listed in Table II. The simulation was run for 76 days, and the results arepresented at 9 days, 27 days and 76 days for a 2-D longitudinal cross-section through theinjection well (Figure 1). The results from the HydroGeoSphere simulation are compared tothose presented in Molson et al. [1992]. Figure 1 shows that the temperature results fromboth models agree. The minor differences that do occur are due to the different treatmentof the temperature boundary condition at the surface of the domain. In addition, Molson etal. [1992] used temperature-dependent density and viscosity terms, whereas the HydroGeo-Sphere simulation treated these parameters as constant under the given range in subsurfacetemperatures.


4.8.1 1D travel time PDF

The travel time PDF for a semi-infinite domain is the flux concentration solution ofEq. (2.153a), which is obtained by applying the boundary conditions gt(t, 0) = δ(t) andD ∂gt(t,x)

∂x |x=∞ = 0, with D = αLv +Dm. This solution reads:

gt(t, x) = x√4πDt t

exp(−(x− vt)2

4Dt

)(4.18)

Considering a 1D domain with outlet position at x = L, the backward travel time PDF isdeduced by replacing x by L− x, to give:

gt(t, x) = L− x√4πDt t

exp(−(L− x− vt)2

4Dt

)(4.19)

In Figure 4.66, the numerical forward and backward travel time PDF’s are compared to thisanalytical solution.


Time(days)

Out

let/I

nlet

Tra

velT

ime

PD

F(1

/day

)

0 20 40 60 80-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Analytic

0 20 40 60 80-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

HydroGeoSphere, backward transport

Figure 4.66: Forward and backward travel time PDF’s versus analytical solution for a 1Dsemi-infinite domain.

Chapter 5

Illustrative Examples


5.1.1 Capture zone probability of a pumping-well

In this section, an illustration of the capture zone probability definition by using the boundaryvalue problem (2.173) is given. The theoretical system corresponds to a uniformly rechargedaquifer (rainfall infiltration of 0.432 m/year), containing a single extraction well (extractionrate of 300 m3/year), and with a natural outlet to which a uniform hydraulic head of 32 m isprescribed (see Fig. 5.1). The aquifer is homogeneous with respect to hydraulic conductivity,porosity and dispersivity. Flow is at steady-state. Figure 5.1 shows the well capture zoneprobability at time t = 1, 5 and 50 days, the date 50 days providing a probability field closeto steady-state (absolute capture zone).

Figure 5.2 shows the solution of Eqs. (2.165) and (2.167). Figure 5.2a provides the averagetime a water particle will take prior to exiting at the pumping-well, exclusively. To do so,Eq. (2.167) has been solved by assigning 〈E〉 = 0 at the well and a zero flux at the naturaloutlet. This solution can be combined with the capture zone probability distribution, forwell protection zone definition purposes.

Figure 5.2b shows the mean age distribution at aquifer scale, figure 5.2c the mean lifeexpectancy distribution, and figure 5.2d gives the mean total transit time (from inlet tooutlets) distribution (〈T 〉 = 〈A〉+ 〈E〉).

5.2 Simulating Tidal Fluctuation

This example shows the use of historical tide data as a boundary condition on a simplesloping surface water domain.

This data was obtained from the Center for Operational Oceanographic Products and

168

CHAPTER 5. ILLUSTRATIVE EXAMPLES 169

Figure 5.1: Pumping-well temporal capture zone probability. The aquifer size is 128× 128×32 m. Iso-probability surfaces 0.1-0.5-0.9.


Figure 5.2: Temporal moment solutions in days: (a) Mean life-expectancy-to-well distribution;(b) Mean age; (c) Mean life expectancy; (d) Mean total transit time.


Services (http://tidesandcurrents.noaa.gov/index.html) for the Station at Redwood City,California. A portion of the raw data file (redwood hl dec12006.orig) is shown here:

Station ID: 9414523 Page Help

Historic Tide DataStation Date Time Vrfy 6DCP#:Units: MetersData%: MSL LST 98.92Maximum: 1.545Minimum: -1.903------- -------- ----- -------9414523 20061201 00:009414523 20061201 00:06...9414523 20061201 02:189414523 20061201 02:24 -1.0339414523 20061201 02:30...

The header contains important information about the data, such as the units (metres) andthe reference elevation (MSL or mean sea level). This data file contains daily minimum andmaximum tide elevations at 6-minute intervals so many of the records do not contain anelevation value. These records are ignored by grok.

This data can be easily modified for use with grok by stripping out the header and footer in-formation and leaving only the data records. This has been done in file redwood hl dec12006.tide data. The beginning of the file is shown here:

9414523 20061201 00:00 0.09414523 20061201 00:06...9414523 20061201 02:189414523 20061201 02:24 -1.0339414523 20061201 02:30...

Note that we set the initial tide elevation value to zero in the first record of the data.

Figure 5.3 shows the surface water domain and finite-element mesh. It is a tilted strip 100m long and 1 m wide, which slopes from z = −2 m at x = 0 to z = 2 m at x = 100 m.

Here is a portion of the main input file, tide.grok, where we assign the tidal boundarycondition:

clear chosen nodes


Figure 5.3: Surface domain and mesh.


choose nodes block0.0 0.00.0 1.0-2.0 -2.0

interpolate specified headtrue

specified head from tidal dataredwood_hl_dec12006.tide_data20061201 00:00

echo flow boundary conditions

output times4.7end

Note that we have chosen to index the time to zero at 20061201 00:00, the first record.

We have chosen to end the simulation at time 4.7 days, as defined by the final (and only)entry to the Output times instruction.

In this problem, time units are in days because we included the instruction Units: kilogram-metre-day.

The resulting head function can be examined by inserting the instruction Echo flow boundaryconditions, which gives the following output:

INSTRUCTION: echo flow boundary conditionsNumber of prescribed head nodes 2

Node Head Time on Time off203 0.0000 0.0000 0.10001

-1.0330 0.10001 0.366671.2350 0.36667 0.66251...

-1.8740 4.8000 0.10000E+21304 0.0000 0.0000 0.10001

-1.0330 0.10001 0.366671.2350 0.36667 0.66251...

-1.5220 0.66251 0.92084-1.8740 4.8000 0.10000E+21

No prescribed flux nodesNo prescribed time-varying flux faces

Figure 5.4 shows the initial water surface (translucent blue) superimposed on the ground


Figure 5.4: Tide level.

surface (green). This figure was produced by Tecplot using the layout file tide.lay. Itconsists of two identical frames which are superimposed. The background frame containsthe ground surface, which is just the finite element mesh shaded green. The foremost frame,representing the water surface, has no background (so it doesn’t obscure the ground surface)and was created by assigning the surface domain head as the z-coordinate and value blankingfor surface water depths less than 1 cm.

An animated movie of the tidal fluctuation can be found in the file tide.avi. Rather thanproduce multiple output files by adding output times, we modified the debug.control fileso that an output file was produced for each timestep.

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Appendix A

Mathematical Notation

The following general rules can be applied throughout this appendix:

• Subscripts i and j are used to denote a property of node i or j respectively.

• Subscript par is used to denote a property of the parent species in the case of a decaychain.

• Symbolˆis used to denote an approximating function e.g. h for approximate hydraulichead.

• Superscript L denotes a time level in a time stepping procedure.

• Superscript r denotes an iteration level in an iterative procedure.

a Subsurface-macropore coupling, distance from block centre to fracture [L].A Tile drain, cross-sectional area in the wetted portion [L2].B Boundary of finite-element volume v [L].Bf Uniform spacing for a set of parallel fractures [L].C Subsurface, solute concentration [M L−3].

Variants:Cd Dual continuum.CImm Double-porosity immobile region.Cf Fracture.Co Surface(overland) flow.Ct Tile drain.CtInj Tile drain, injected water.Cw Well.CwInj Well, injected water.Cc Channel.

181

APPENDIX A. MATHEMATICAL NOTATION 182

CcInj Channel, injected water.Ch Chezy coefficient [L1/2 T−1].Cdwn Concentration of downstream node between i and j [M L−3].

Variants:Cups Upstream node.Ci2ups Second upstream node.

CL A constant which depends on rainfall intensity r [-].Cx Chezy coefficient in the x−direction [L1/2 T−1].Cy Chezy coefficient in the y−direction [L1/2 T−1].D Subsurface, hydrodynamic dispersion tensor [L2 T−1].

Variants:Dd Dual continuum.Df Fracture.Do Surface(overland) flow.

Dfree Solute, free-solution diffusion coefficient [L2 T−1].D∗Imm Double-porosity, effective diffusion coefficient in the immobile region [L2 T−1].do Surface(overland) flow, water depth [L].Dt Tile drain, dispersion coefficient [L2 T−1].Dw Well, dispersion coefficient [L2 T−1].ETS Surface water, evapotranspiration [L3 T−1].ETG Subsurface water, evapotranspiration [L3 T−1].F Jacobian matrix.fs Fracture, spacing [L].fx Darcy-Weisbach friction factor in the x−direction [-].fy Darcy-Weisbach friction factor in the y−direction [-].g Gravitational acceleration [L T−2].h Subsurface, hydraulic head [L].

Variants:hd Dual continuum.hf Fracture.ho Surface(overland) flow, water surface elevation.ht Tile drain.hw Well.

Hd Surface(overland) flow, depression storage height [L].Ho Surface(overland) flow, obstruction storage height [L].Hs Surface(overland) flow, maximum height over which area covered by surface

water goes from 0 to unity [L].I Net infiltration [L3 T−1].ITmax The maximum number of iterations allowed during a single time level.I Identity tensor.k Subsurface, permeability tensor [L2].K Subsurface, saturated hydraulic conductivity tensor [L T−1].K∗ Component of hydraulic conductivity tensor normal to a seepage face.K ′ Subsurface, equilibrium distribution coefficient [M−1 L3].

Variants:


K ′d Dual continuum.K ′f Fracture.

Ka Subsurface-macropore coupling, interface hydraulic conductivity [L T−1].kd Dual continuum, permeability tensor [L2].Kd Dual continuum, saturated hydraulic conductivity tensor [L T−1].Kf Fracture, saturated hydraulic conductivity tensor[L T−1].Ko Surface(overland) flow, conductance tensor[L T−1].kr Subsurface, relative permeability [-].

Variants:krd Dual continuum.krf Fracture.kro Surface(overland) flow.krt Tile drain.krw Well.

kra Subsurface-macropore coupling, interface relative permeability[-].

krso Subsurface-surface(overland) flow coupling, rill effect term [-].K Bulk modulus of porous media [M T−2 L−1].Kf Bulk modulus of fluid [M T−2 L−1].Kss Bulk modulus of solids [M T−2 L−1].Ks Surface flow, conductance term reduction factor [-].Kso Subsurface-surface (overland) flow coupling, leakance term [T−1].Kt Tile drain, hydraulic conductivity [L T−1].Kw Well, saturated hydraulic conductivity [L T−1].Kox Surface(overland) flow, conductance in x− direction [L T−1].Koy Surface(overland) flow, conductance in y− direction [L T−1].l Well or tile drain, length coordinate along the axis [L].l′ Location at which specified well or tile discharge (or recharge) is applied [L].lp Pore-connectivity parameter for unsaturated functions [-].Ls Well, screen length [L].N Finite element basis function [-].n Manning roughness coefficient [T L−1/3].n∗ Brooks-Corey exponent equal to 2 + 3λ∗ [-].nx Manning roughness coefficient in the x−direction [L−1/3 T].ny Manning roughness coefficient in the y−direction [L−1/3 T].P Net precipitation [L3 T−1].q Subsurface, fluid flux [L T−1].

Variants:qd Dual continuum.qf Fracture.qo Surface(overland) flow.qt Tile drain.qw Well.

Pc Capillary pressure head [L].Pdwn Position vector of downstream node between i and j.


Variants:Pups Upstream node.Pi2ups Second upstream node.

Pw Well, wetted perimeter [L].Pt Tile drain wetted perimeter [L].Q Subsurface, fluid source or sink [T−1].Qc Channel, fluid source or sink [L3 T−1].Qcd Dual continuum, solute source or sink [T−1].Qd Dual continuum, fluid source or sink [T−1].QWG Subsurface water, withdrawal [L3 T−1].QG1 Subsurface water, inflow [L3 T−1].QG2 Subsurface water, outflow [L3 T−1].QGS Surface/subsurface water interactive flow [L3 T−1].Qo Surface(overland) flow, volumetric flow rate per unit area representing external

source and sinks [L T−1].QWS Surface water, withdrawal [L3 T−1].QS1 Surface water, inflow [L3 T−1].QS2 Surface water, outflow [L3 T−1].Qt Tile drain, specified fluid flow rate in or out [L3 T−1], applied at location l′.Qw Well, discharge (or recharge) per unit length [L2 T−1] applied at location l′.r Rainfall intensity [L T−1].R Subsurface, retardation factor [-].

Variants:Rd Dual continuum.Rf Fracture.Ro Overland flow.Rt Tile drain.

r0 Double porosity, radius of a representative sphere [L].rc Well, casing radius [L].rs Well, screen radius [L].Rei Reynolds number in coordinate direction i.s Surface(overland) flow, coordinate along direction of maximum ground surface

slope [L].Se Effective saturation [-].Sfx Surface(overland) flow, friction slope in the x−direction [-].Sfy Surface(overland) flow, friction slope in the y−direction [-].Sox Surface(overland) flow, bed slope in the x−direction [-].Soy Surface(overland) flow, bed slope in the y−direction [-].Ss Subsurface, specific storage [L−1].Ssf Fracture, specific storage [L−1].Sw Subsurface, water saturation [-].

Variants:Swd Dual continuum.Swf Fracture.Swt Tile drain.


Sww Well.Swmax The maximum change in water saturation allowed during a single time-step.Swr Residual water saturation [-].t Time [T].tolb, tolf Switching parameters for primary variable substitution [-].v Region or control volume associated with a node [L3].V Volume of the finite-element domain [L3].vio Surface(overland) flow, vertically averaged flow velocity in the coordinate

direction i [L T−1].vxo Surface(overland) flow, vertically averaged flow velocity in the x−direction

[L T−1].vyo Surface(overland) flow, vertically averaged flow velocity in the y−direction

[L T−1].wd Dual continuum, volumetric fraction of the total porosity [-].wf Fracture, aperture or width [L].wm Subsurface, volumetric fraction of the total porosity [-].W A large number e.g. 1020.x, y, z Global Cartesian coordinates [L].xi Component of the Cartesian coordinate system [L].z Subsurface, elevation head [L].

Variants:zd Dual continuum.zf Fracture.zt Tile drain.zw Well.

z′ Depth coordinate from the soil surface [L].zo Overland flow, bed (land surface) elevation [L].α Van Genuchten parameter [L−1].αImm Double-porosity, first-order mass transfer coefficient between the mobile

and immobile regions [T−1].αl Subsurface, longitudinal dispersivity [L].

Variants:αld Dual continuum.

αs Subsurface-macropore coupling, first-order mass transfer [T−1].αt Subsurface, transverse dispersivity [L].

Variants:αtd Dual continuum.

αw Water, compressibility [L T2 M−1].αm Matrix compressibility [L T2 M−1].αss Solids compressibility [L T2 M−1].αwd Subsurface-macropore coupling, first-order fluid exchange coefficient [L−1 T−1].α∗wd Subsurface-macropore coupling, geometric factor [L−2].α′ = 1−K/Ks [-]β Van Genuchten parameter [-].βd Subsurface-macropore coupling, geometrical factor [-].


βske Skempton’s coefficient [-]γ Water, kinematic viscosity [L2 T−1].γij Term describing fluid flow between nodes i and j. [L2 T−1]Γex Subsurface, fluid exchange rate with all other domains [T−1].Γd Dual continuum, fluid exchange rate with subsurface domain [T−1].

Variants:Γf Fracture.Γo Surface(overland) flow.Γt Tile drain.Γw Well.

γw Subsurface-macropore coupling, empirical constant [-].δ Dirac delta function.∆SG Subsurface, change in water storage [L3].∆SS Surface flow, change in water storage [L3].∆t Time step [T].ε A small numerical shift in the pressure head value used in the Newton-

Raphson method.ηi Subsurface, a set of nodes connected to node i.

Variants:ηdi Dual continuum.ηf i Fracture.ηwi Well.ηti Tile drain.

θ Subsurface, water content [-].θImm Double-porosity, porosity of the immobile region [-].θs Subsurface, saturated water content [-].

Variants:θsd Dual continuum.

λ Subsurface, solute first-order decay constant [L−1].Variants:

λd Dual continuum.λf Fracture.λo Surface(overland) flow.λt Tile drain.λw Well.

λ∗ Brooks-Corey pore-size index [-].µ Water, viscosity [M L−1 T−1].ν Van Genuchten parameter equal to 1− 1

β [-].ν∗ Poisson’s ratio [-].π The constant 3.1415· · · . Would you prefer apple or cherry? Ice cream

with that?ψ Subsurface, pressure head [L].

Variants:ψd Dual continuum.ψf Fracture.


ψt Tile drain.ψw Well.

ψatm Atmospheric pressure [L].ψb An assigned pressure head [L].ρ Water, density [M L−3].ρb Subsurface, bulk density [M L−3].

Variants:ρbd Dual continuum.

σ(r) Van Leer flux limiter [-] with smoothness sensor r.σzz Surface vertical stress [M T−2 L−1 ]τ Subsurface, matrix tortuosity [-].

Variants:τd Dual continuum.

φo Surface(overland) flow, surface porosity [-].χ Water, surface tension [M T−2].Ωex Subsurface, solute exchange rate with all other domains [T−1].Ωd Dual continuum, solute exchange rate with subsurface domain [T−1].

Variants:ΩImm Double-porosity immobile zone.Ωf Fracture.Ωo Surface(overland) flow.Ωt Tile drain.Ωw Well.

∇ One-dimensional gradient operator.∇ Two-dimensional gradient operator.∇ Three-dimensional gradient operator.ζ Loading efficiency [-].

Index

Aquifer, see Porous mediumAquitard, see Porous medium

Boundary conditionsflow

Evapotranspiration, 28, 69

Channelsnumerical techniques

transport, 74theory

transport, 38Control volume finite-element method

numerical techniques, 56

Decay, see Radioactive decayDiscrete fractures

numerical techniquesflow, 60transport, 72

theoryflow, 12transport, 35

Double porosity transportcoupling, 45numerical techniques, 73theory, 36

Dual continuumnumerical techniques

flow, 61flow coupling , 65transport, 73

theoryflow, 13flow coupling , 27

transport, 36transport coupling , 45

Elemental velocities, 70

Finite difference solutiontheory, 55

Fractures, see Discrete fractures

Hydromechanical couplingtheory

flow, 14

Illustrative examplesflow

Tidal fluctuation, 168transport

Capture zone probability of apumping well, 168

Isotopic fractionationtransport coupling, 47

Isotopic fractionation transportnumerical techniques, 73theory, 36

Mass balancenumerical techniques, 82

Matrix solvernumerical techniques, 79

Newton-Raphson methodnumerical techniques, 79

Porous mediumnumerical techniques

flow, 59

188

INDEX 189

transport, 71theory

flow, 9transport, 34

Primary variable substitutionnumerical techniques, 81

Radioactive decayexample, 125, 128theory, 34

Rill storagetheory, 20

Runoffnumerical techniques

flow, 61transport, 74

theoryflow, 17transport, 38

Solution proceduresnumerical techniques, 83

Subsurface domainnumerical techniques

elemental velocities, 70flow, 59flow boundary conditions, 67flow coupling, 64transport boundary conditions, 78transport coupling , 76travel time probability , 75

theoryflow boundary conditions, 28transport boundary conditions, 47transport coupling , 45–47travel time probability, 47

Surface domainboundary conditions

flow, 28, 68transport, 47

numerical techniquesflow, 61

numerical techniquesflow coupling , 65

theoryrill storage, 20

coupling, 27

Thermal energynumerical techniques

transport, 75transport coupling, 46

Tile drainsnumerical techniques

transport, 74theory

transport, 38Time stepping, 2Time stepping

adaptivenumerical techniques, 82

Verification examplessubsurface flow

drainage of a fractured tuff column,Wang and Narasimhan, 88

dry initial conditions, Forsyth, 86hydromechanical coupling, 94hydromechanical coupling with

external stresses, 94Theis solution, 84unsaturated flow through a column,

Huyakorn, 85surface flow

1-D surface flow [Govindaraju et al.,1988a and 1988b], 96

2-D surface flow[diGiammarco etal., 1996], 111

surface/subsurface flow3-D field scale study [Abdul, 1985],

1203-D field scale study [Abdul, 1985]),

125conjunctive surface-subsurface flow

[Smith and Woolhiser 1971], 101transport

1-D travel time PDF, 166chain decay in a fracture, 128chain decay in a porous medium,

125dual-continuum, 133dual-porosity , 133

INDEX 190

heat transfer in fractured media,155

heat transfer in porous media, 155injection/Withdrawal Well , 134injection/Withdrawal Well Pair,

140injection/withdrawal well pair in an

ambient flow field, 142plume in a heterogeneous aquifer,

145point source, uniform, steady flow ,

137

time-varying source, 130unsaturated soil slab, 146variable-density flow in fractured

porous media, 152variable-density flow in porous

media, 150

Wellsnumerical techniques

transport, 73theory

transport, 37

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